Chapter 17: Functions of Several Variables (Set-5)
For f(x,y)=x2yx4+y2f(x,y)=x4+y2x2y, the limit at (0,0)(0,0) is
A Does not exist
B 0
C 1
D Infinity
Along y=0y=0, f=0f=0. Along y=x2y=x2, f=x4x4+x4=12f=x4+x4x4=21. Since two paths give different values, the limit is not unique, so it does not exist.
For f(x,y)=x3x2+y2f(x,y)=x2+y2x3, as (x,y)→(0,0)(x,y)→(0,0) the limit
A Does not exist
B Equals 1
C Equals 0
D Equals infinity
Use polar: f=r3cos3θr2=rcos3θ→0f=r2r3cos3θ=rcos3θ→0 as r→0r→0. Since the limit becomes 0 independent of θθ, the limit exists and equals 0.
If f(x,y)=x2−y2x2+y2f(x,y)=x2+y2x2−y2, then as (x,y)→(0,0)(x,y)→(0,0)
A Limit does not exist
B Limit equals 1
C Limit equals infinity
D Limit equals 0
In polar, numerator =r2(cos2θ−sin2θ)=r2cos2θ=r2(cos2θ−sin2θ)=r2cos2θ. Denominator =r=r. So f=rcos2θ→0f=rcos2θ→0 as r→0r→0, independent of angle.
For f(x,y)=xyx2+y2f(x,y)=x2+y2xy, the limit at (0,0)(0,0) is
A Does not exist
B 0
C 1/2
D 1
Along y=xy=x, f=x22x2=1/2f=2x2x2=1/2. Along y=−xy=−x, f=−x22x2=−1/2f=2×2−x2=−1/2. Different path values imply the limit does not exist.
If lim(x,y)→(0,0)f(x,y)=Llim(x,y)→(0,0)f(x,y)=L exists, then limit along curve y=x2y=x2 must
A Be 0 always
B Equal LL
C Be 1 always
D Not exist
Existence of the full multivariable limit forces agreement along every path, including nonlinear curves like y=x2y=x2. Any mismatch on even one curve disproves the limit.
A function can have all directional derivatives at origin but still
A Be constant
B Be polynomial
C Be discontinuous
D Be linear
Directional derivatives test behavior along lines, but discontinuity can still occur due to nonlinear path behavior. So having all directional derivatives does not guarantee continuity or differentiability.
A sufficient condition for continuity of f(x,y)f(x,y) at a point is
A It is polynomial there
B Mixed partials exist
C Hessian is nonzero
D Jacobian is nonzero
Polynomials are continuous everywhere. This provides an immediate sufficient condition. Other listed conditions are not direct continuity guarantees for a general function.
If ff and gg are continuous at (a,b)(a,b) and g(a,b)≠0g(a,b)=0, then f/gf/g is
A Discontinuous there
B Always undefined
C Continuous only on axes
D Continuous there
Quotient of continuous functions is continuous wherever the denominator is nonzero. Since g(a,b)≠0g(a,b)=0, there is a neighborhood where g≠0g=0, ensuring continuity of f/gf/g.
For f(x,y)=∣xy∣f(x,y)=∣xy∣, at (0,0)(0,0) the partial derivatives fx,fyfx,fy are
A Both 1
B Do not exist
C Both 0
D Infinity
fx(0,0)=limh→0∣h⋅0∣−0h=0fx(0,0)=limh→0h∣h⋅0∣−0=0. Similarly fy(0,0)=0fy(0,0)=0. Partial derivatives exist and are zero, though differentiability needs more checking.
For f(x,y)=∣xy∣f(x,y)=∣xy∣, differentiability at (0,0)(0,0) is
A True
B False
C Depends on path
D Undefined statement
∣xy∣≤x2+y22∣xy∣≤2×2+y2. Hence ∣xy∣x2+y2≤x2+y22→0x2+y2∣xy∣≤2×2+y2→0. This shows f=o(r)f=o(r), so differentiable at origin with gradient 0.
If fxfx and fyfy exist at a point, then ff is guaranteed
A Continuous always
B Not guaranteed continuous
C Differentiable always
D Linear always
Existence of partial derivatives at a point does not ensure continuity. There are standard counterexamples where both partials exist at origin but the function is discontinuous due to path issues.
If fx,fyfx,fy exist near (a,b)(a,b) and are continuous at (a,b)(a,b), then ff is
A Discontinuous at (a,b)(a,b)
B Not defined at (a,b)(a,b)
C Differentiable at (a,b)(a,b)
D Non-invertible there
Continuity of first partial derivatives in a neighborhood is a standard sufficient condition for differentiability. It ensures a good linear approximation with error smaller than the distance to the point.
For z=f(x,y)z=f(x,y), maximum directional derivative at a point equals
A ∥∇f∥∥∇f∥
B fx+fyfx+fy
C ∥Hessian∥∥Hessian∥
D fxyfxy
Directional derivative in unit direction uu is ∇f⋅u∇f⋅u. Its maximum over unit vectors occurs when uu aligns with ∇f∇f, giving value ∥∇f∥∥∇f∥.
A unit direction giving maximum increase of ff at a point is
A Any tangent vector
B Any normal vector
C Zero vector
D ∇f/∥∇f∥∇f/∥∇f∥
The steepest ascent direction is the normalized gradient. Dividing by its magnitude gives a unit vector, ensuring the directional derivative equals the maximum possible rate of increase.
For implicit surface F(x,y,z)=0F(x,y,z)=0, a normal direction is
A ∇f∇f
B ⟨x,y,z⟩⟨x,y,z⟩
C ∇F∇F
D Any tangent vector
The gradient of FF is perpendicular to level surfaces F=constantF=constant. So on F=0F=0, ∇F∇F provides a normal direction whenever ∇F≠0∇F=0.
If ff is homogeneous degree nn, then fxfx is homogeneous degree
A n−1n−1
B nn
C n+1n+1
D 0
Differentiation reduces degree by 1 for homogeneous functions. Scaling (x,y)→(tx,ty)(x,y)→(tx,ty) shows fx(tx,ty)=tn−1fx(x,y)fx(tx,ty)=tn−1fx(x,y), matching degree n−1n−1.
If ff is homogeneous degree nn, then xfx+yfyxfx+yfy equals
A ff
B nfnf
C 0
D n−1n−1
This is Euler’s theorem: xfx+yfy=nfxfx+yfy=nf. It provides a quick identity that connects the function and its first partial derivatives for homogeneous functions.
If ff is homogeneous degree n≠0n=0, then f=1n(xfx+yfy)f=n1(xfx+yfy) is
A False always
B Only for polynomials
C True identity
D Only for rationals
Rearranging Euler’s theorem gives f=1n(xfx+yfy)f=n1(xfx+yfy) when n≠0n=0. This is often used to simplify expressions involving ff and its partial derivatives.
To check homogeneity of f(x,y)=x2+y2f(x,y)=x2+y2, degree is
A 2
B 0
C 1/2
D 1
f(tx,ty)=t2(x2+y2)=∣t∣x2+y2f(tx,ty)=t2(x2+y2)=∣t∣x2+y2. For t>0t>0, this is tf(x,y)tf(x,y), so it is homogeneous of degree 1 in the usual scaling sense.
For f(x,y)=ln(x2+y2)f(x,y)=ln(x2+y2), the function is homogeneous
A Not homogeneous
B Degree 2
C Degree 0
D Degree 1
f(tx,ty)=ln(t2(x2+y2))=ln(t2)+ln(x2+y2)f(tx,ty)=ln(t2(x2+y2))=ln(t2)+ln(x2+y2), which is not tnf(x,y)tnf(x,y). The extra additive term breaks homogeneity.
If u=u(x,y),v=v(x,y)u=u(x,y),v=v(x,y) and mapping is locally invertible, then Jacobian must be
A Zero
B Always 1
C Nonzero
D Always negative
Local invertibility requires the derivative matrix to be invertible, which needs a nonzero determinant. Hence ∂(u,v)/∂(x,y)≠0∂(u,v)/∂(x,y)=0 at that point is essential.
For composite mapping (x,y)→(r,θ)→(u,v)(x,y)→(r,θ)→(u,v), Jacobian satisfies
A Multiply determinants
B Add determinants
C Subtract determinants
D Divide determinants
Jacobians follow chain rule: ∂(u,v)/∂(x,y)=∂(u,v)/∂(r,θ)⋅∂(r,θ)/∂(x,y)∂(u,v)/∂(x,y)=∂(u,v)/∂(r,θ)⋅∂(r,θ)/∂(x,y). This extends the single-variable derivative chain rule.
If J=∂(u,v)∂(x,y)J=∂(x,y)∂(u,v) is negative, it indicates
A Limit fails
B Orientation reversed
C Continuity fails
D Differentiability fails
The sign of Jacobian shows orientation change: negative means the mapping flips orientation locally. It does not automatically imply discontinuity or non-existence of derivatives.
For polar change, ∂(r,θ)∂(x,y)∂(x,y)∂(r,θ) equals
A rr
B r2r2
C 11
D 1/r1/r
Since ∣∂(x,y)/∂(r,θ)∣=r∣∂(x,y)/∂(r,θ)∣=r, the inverse Jacobian magnitude is ∣∂(r,θ)/∂(x,y)∣=1/r∣∂(r,θ)/∂(x,y)∣=1/r where r≠0r=0, consistent with reciprocal relation.
If u=x2−y2u=x2−y2, v=2xyv=2xy, then ∂(u,v)∂(x,y)∂(x,y)∂(u,v) equals
If u=x+y+zu=x+y+z, v=x−yv=x−y, w=zw=z, then ∂(u,v,w)∂(x,y,z)∂(x,y,z)∂(u,v,w) is
A 2
B 0
C −2
D 1
Matrix is (1111−10001)1101−10101. Determinant equals determinant of (111−1)(111−1) times 1, giving −2−2.
For spherical coordinates, Jacobian factor ρ2sinϕρ2sinϕ becomes zero when
A ϕ=0,πϕ=0,π
B θ=0θ=0 only
C ρ=1ρ=1 only
D ϕ=π/2ϕ=π/2 only
sinϕ=0sinϕ=0 at ϕ=0ϕ=0 and ϕ=πϕ=π, so the Jacobian vanishes on the polar axis. This reflects degeneracy of spherical coordinates at the poles.
Functional dependence of u,v,wu,v,w may be indicated by
A Jacobian equals one
B Jacobian negative
C Jacobian constant
D Jacobian equals zero
If u,v,wu,v,w are functionally dependent, the rank drops and the Jacobian determinant ∂(u,v,w)/∂(x,y,z)∂(u,v,w)/∂(x,y,z) can become zero in the relevant region, signaling dependence.
If D=fxxfyy−(fxy)2<0D=fxxfyy−(fxy)2<0 at critical point, then point is
A Local minimum
B Local maximum
C Saddle point
D Global minimum
Negative DD means curvature differs by direction: upward in one direction and downward in another. That is the defining behavior of a saddle point in two-variable functions.
• If D>0 at a critical point, then fxx and fyy must
A Have same sign
B Have opposite sign
C Both be zero
D Be undefined
If D>0D>0, then fxxfyy>(fxy)2≥0fxxfyy>(fxy)2≥0, so fxxfyy>0fxxfyy>0. Hence fxxfxx and fyyfyy must have the same sign (both positive or both negative.
If D=0D=0 at critical point, then classification needs
A Only Jacobian
B Higher-order terms
C Only Euler theorem
D Only substitution
When D=0D=0, the quadratic approximation is not decisive. One must examine higher-degree terms in the Taylor expansion or use a different local analysis method.
A necessary condition for constrained extremum of ff with constraint g=cg=c is
A ∇f∥∇g∇f∥∇g
B ∇f⊥∇g∇f⊥∇g
C f=gf=g
D g=0g=0 always
At constrained optimum, moving along the constraint should not change ff first-order. Thus the gradient of ff must be parallel to gradient of gg, giving ∇f=λ∇g∇f=λ∇g.
If ∇g=0∇g=0 at a constraint point, Lagrange method becomes
A Always works
B Gives unique answer
C May fail
D Same as unconstrained
Lagrange multipliers rely on ∇g≠0∇g=0 to define a valid normal to the constraint. If ∇g=0∇g=0, the constraint may be singular and the method may miss solutions.
For f(x,y)=x2−y2f(x,y)=x2−y2, the critical point at (0,0)(0,0) is
A Local minimum
B Local maximum
C No critical point
D Saddle point
fx=2xfx=2x, fy=−2yfy=−2y, so critical point at origin. fxx=2fxx=2, fyy=−2fyy=−2, fxy=0fxy=0. D=2(−2)−0=−4<0D=2(−2)−0=−4<0, hence saddle.
For f(x,y)=x2+y2f(x,y)=x2+y2, the point (0,0)(0,0) is
A Global minimum
B Saddle point
C Global maximum
D Not critical
f≥0f≥0 for all (x,y)(x,y) and equals 0 at origin. Also fx=2x,fy=2yfx=2x,fy=2y so origin is critical. Both curvature directions are upward, giving a minimum (in fact global).
A classic example where partials exist at origin but function not continuous is
A x2+y2x2+y2
B x2yx2y
C xyx2+y2x2+y2xy with 0
D sin(x+y)sin(x+y)
Define f(0,0)=0f(0,0)=0. Along axes, value is 0 so partials exist. Along y=xy=x, value tends to 1/21/2. Hence function is not continuous at origin despite existing partials.
If all first partials exist and are continuous near a point, then ff is
A Differentiable there
B Discontinuous there
C Not defined there
D Non-invertible there
Continuity of first partial derivatives in a neighborhood is a sufficient condition for differentiability. It guarantees the linear approximation error becomes negligible compared to distance to the point.
If ff is differentiable at a point, then all directional derivatives
A Fail there
B Exist there
C Equal 1
D Are infinite
Differentiability implies a linear map approximation, so directional derivatives exist in every direction and are given by ∇f⋅u∇f⋅u. This is one key consequence of differentiability.
A directional derivative can exist in every direction but ff still not differentiable because
A Gradient must be zero
B Function must be bounded
C Jacobian must be 1
D Linear approximation fails
Differentiability requires a single linear map that approximates ff in all directions with small error. Directional derivatives alone do not guarantee this uniform linear behavior.
For f(x,y)=x2+y2f(x,y)=x2+y2, the gradient at origin is
A Does not exist
B ⟨0,0⟩⟨0,0⟩
C ⟨1,1⟩⟨1,1⟩
D Infinity
Although ff is continuous at origin, the direction of steepest change depends on approach and the function is not differentiable there. Hence the gradient is not defined at (0,0)(0,0).
If f(x,y)f(x,y) is continuous and g(x,y)→0g(x,y)→0, then f(x,y)g(x,y)→0f(x,y)g(x,y)→0 provided
A ff is differentiable
B ff is homogeneous
C ff is bounded near point
D Jacobian is nonzero
If ff stays bounded near the point and g→0g→0, then ∣fg∣≤M∣g∣→0∣fg∣≤M∣g∣→0. Continuity at the point usually ensures boundedness in a neighborhood.
In R2R2, a common norm used in epsilon–delta is
A Euclidean norm
B Only x-distance
C Only y-distance
D Manhattan always required
The standard distance is (x−a)2+(y−b)2(x−a)2+(y−b)2. Using this norm, continuity and limits are defined. Other norms are possible but Euclidean is most common in calculus.
If ff is homogeneous degree nn, then ∇f⋅⟨x,y⟩∇f⋅⟨x,y⟩ equals
A ff
B 0
C n−1n−1
D nfnf
∇f⋅⟨x,y⟩=xfx+yfy∇f⋅⟨x,y⟩=xfx+yfy. By Euler’s theorem for homogeneous functions of degree nn, this equals nf(x,y)nf(x,y).
For u=x2+y2u=x2+y2 and v=x/yv=x/y, the mapping fails where
A x=0x=0
B y=0y=0
C x=yx=y
D x2+y2=1×2+y2=1
v=x/yv=x/y is undefined when y=0y=0. Even if uu is defined everywhere, the overall mapping (x,y)↦(u,v)(x,y)↦(u,v) is not defined along the line y=0y=0.
A Jacobian used for change of variables in integrals must be taken as
A Absolute value
B Negative value only
C Always squared
D Always ignored
In integration, the measure must be nonnegative, so we use ∣J∣∣J∣. The sign of Jacobian indicates orientation, but area/volume scaling in integrals uses its magnitude.
If f(x,y)=0f(x,y)=0 for all points on a neighborhood except one point, then ff is continuous at that point if
A Value set to 1
B Value set to infinity
C Value set to 0
D Value left undefined
If ff is zero in a punctured neighborhood, the limit at the point is 0. Continuity requires defining ff at that point as 0 so the value matches the limit.
In second derivative test, condition D>0D>0 but fyyfyy sign differs from fxxfxx
A Cannot happen
B Always saddle
C Always minimum
D Always maximum
If D=fxxfyy−(fxy)2>0D=fxxfyy−(fxy)2>0, then fxxfyy>(fxy)2≥0fxxfyy>(fxy)2≥0, so fxxfyy>0fxxfyy>0. Hence fxxfxx and fyyfyy must have the same sign.
For f(x,y)=x2y2x2+y2f(x,y)=x2+y2x2y2, the limit at origin is
A 1
B Does not exist
C Infinity
D 0
In polar: numerator =r4cos2θsin2θ=r4cos2θsin2θ, denominator =r2=r2. So f=r2cos2θsin2θ→0f=r2cos2θsin2θ→0 as r→0r→0, independent of θθ.
If ∣f(x,y)∣≤Cx2+y2∣f(x,y)∣≤Cx2+y2 near origin, then ff is
A Discontinuous always
B Not defined
C Continuous at origin
D Infinite at origin
Since x2+y2→0x2+y2→0, the inequality gives ∣f(x,y)∣→0∣f(x,y)∣→0. If f(0,0)f(0,0) is defined as 0, then the limit equals the value, giving continuity at origin.
A key reason Jacobians matter in multivariable limits is that they can
A Guarantee continuity
B Simplify by substitution
C Force differentiability
D Replace epsilon–delta
Coordinate substitutions (polar, cylindrical, spherical) often simplify expressions near a point. Jacobians describe how measures change, and the substitution can reveal limit behavior through simpler variables like rr.