Practice Questions on Functions of Several Variables with Answers

**Questions on Functions of Several Variables with Answers**

**Q1. A function f(x,y) on R ^{2} has the following partial derivatives**

^{∂f(x,y)}/_{∂x} = x^{2} , ^{∂f(x,y)}/_{∂y} = y^{2}

**Then**

(a) f has directional derivatives in all directions everywhere

(b) f has derivative at all points

(c) f has directional derivative only along the direction (1,1) everywhere

(d) f does not have directional derivatives in any direction everywhere

**Q2. A function f:R ^{2} → R is defined by f(x,y) = xy. Let v = (1,2) and a = (a_{1}, a_{2}) be two elements of R^{2}. The directional derivative of f in the direction of v at a is**

(a) a_{1} + 2a_{2}

(b) a_{2} + 2a_{1}

(c) a_{2}/2 + a_{1}

(d) a_{1}/2 + a_{2}

**Q3. Let f:R ^{2} → R^{2} be given by f(r,θ) = (rcosθ , rsinθ). Then for which of the subset U of R^{2} given below, f restricted to U admits an inverse? **

(a) U = R^{2}

(b) U = {(x,y)∈R^{2} : x > 0 , y > 0}

(c) U = {(x,y)∈R^{2} : x^{2} + y^{2} < 1}

(d) U = {(x,y)∈R^{2} : x < -1 , y < -1}

**Q4. Consider the function f(x,y) = x ^{2}/y^{2} , (x,y) ∈ [^{1}/_{2},^{3}/_{2}]×[^{1}/_{2},^{3}/_{2}]. Then derivative of the function at (1,1) along the direction (1,1)**

(a) 0

(b) 1

(c) 2

(d) -2

**Q5. Let ****f(x,y)**** = ^{xy}/(x^{2} + y^{2}) ,(x,y) ≠ (0,0) ; f(0,0) = 0 . Then**

(a) f is continuous at (0,0) and the partial derivatives *f _{x’}f_{y}* exist at every point of R

^{2}

(b) f is discontinuous at (0,0) and

*f*f exist at every point on R

_{x’}^{2}

(c)

*f*is discontinuous at (0,0) and

*f*exist only at (0,0)

_{x’}f(d) None of these

**Q6. Let f:R ^{2} → R and g:R^{2} → R be defined by f(x,y) = |x| + |y| and g(x,y) = |xy| . Then**

(a) f is differentiable at (0,0), but g is not differentiable at (0,0)

(b) f is differentiable at (0,0), but f is not differentiable at (0,0)

(c) Both f and g are differentiable at (0,0)

(d) Both f and g are continuous at (0,0)

**Q7. Consider the map f:R ^{2} → R^{2} defined by f(x,y) = (3x – 2y + x^{2}, 4x+5y+y^{2}). Then**

(a) f us continuous at (0,0)

(b) f is continuous at (0,0) and all directional derivatives exists at (0,0)

(c) f is differentiable at (0,0) but the derivative Df(0,0) is not invertible

(d) f is differentiable at (0,0) but the derivative Df(0,0) is invertible

**Q8. The map L:R ^{2} → R^{2} given by L(x,y) = (x-y) is**

(a) Differentiable everywhere on R^{2}

(b) Differentiable only at (0,0)

(c) DL(0,0) = L

(d) DL(x,y) = L for all (x,y) ∈ R^{2}

**Q9. Let f:[π, 2π] → R ^{2} be the function f(t) = (cost, sint) . Which of the following are necessarily correct ?**

(a) There exists t_{o} ∈ [π, 2π] such that f'(t_{o}) = ^{1}/_{π} [f(2π) – f(π)]

(b) There does not exist any t_{o} ∈ [π, 2π] such that f'(t_{o}) = ^{1}/_{π} [f(2π) – f(π)]

(c) There exists t_{o} ∈ [π, 2π] such that || f(2π) – f(π) || ≤ || f'(t_{o}) ||

(d) f'(t) = (-sint, cost) for all t ∈ [2π, π]

**Q10. Let f:R ^{n} → R^{n} be the function defined by f(x) = x||x||^{2} for x ∈ R^{n} . Which of the following statements are correct ?**

(a) (Df)(0) = 0

(b) (Df)(x) = 0 for all x ∈ R^{n}

(c) f is one-one

(d) f has an inverse

**Q11. Let f:A∪E → R ^{2} be differentiable, where**

**A = {(x,y) ∈ R ^{2} : ^{1}/_{2} < x^{2} < y^{2} < 1} and**

**E = {(x,y) ∈ R**

^{2}: (x-2)^{2}+ (y-2)^{2}<^{1}/_{2}} .**Let Df be the derivative of the function f . Which of the following are necessarily correct ?**

(a) If (Df)(x,y) = 0 for all (x,y) ∈ A∪E, then f is constant

(b) If (Df)(x,y) = 0 for all (x,y) ∈ A, then f is constant on A

(c) If (Df)(x,y) = 0 for all (x,y) ∈ E, then f is constant on E

(d) If (Df)(x,y) = 0 for all (x,y) ∈ A∪E, then for some (x_{o, }y_{o}), (x_{1}, x_{2}) ∈ R^{2},

f(x,y) = (x_{o}, y_{o}) for all (x,y) ∈ A and

f(x,y) = (x_{1}, y_{1}) for all (x,y) ∈ E

**Q12. L:R ^{n} → R be the function L(x) = <x, y>, where <•,•> is some inner product on R^{n} and y is a fixed vector in R^{n}. Further denote by DL, the derivative of L. Which of the following are necessarily correct ?**

(a) DL(u) = DL(v) for all u,v ∈ R^{n}

(b) DL(0,0,…,0) = L

(c) DL(x) = ||x||^{2} for all x ∈ R^{n}

(d) DL(1,1,…,1) = 0

**Q13. Let A = {(x,y) ∈ R ^{2} : x+y ≠ -1} define f:A → R^{2} by f(x,y) = [^{x}/_{(1+x+y)} , ^{y}/_{(1+x+y)}]. Then**

(a) The Jacobian matrix of f does not vanish on A

(b) f is infinitely differentiable on A

(c) f is injective on A

(d) f(A) = R^{2}

**Q14. Let f:R ^{n} → R be the map f(x_{1},…,x_{n}) = a_{1}x_{1} + …+a_{n}x_{n} , where a = (a_{1},…a_{n}) is a fixed non-zero vector. Let Df(0) denote the derivative of f at 0. Which of the following are correct ?**

(a) (Df)(0) is a linear map from Rn to R

(b) [(Df)(0)](a) = ||a||^{2}

(c) [(Df)(0)](a) = 0

(d) [(Df)(0)](b) = a_{1}b_{1}+…+anbn for b = (b_{1},…,b_{n})

**Q15. Let f(x,y) = √(|xy|) . Then**

(a) f_{x} and f_{y} do not exist at (0,0)

(b) f_{x}(0,0) = 1

(c) f_{y}(0,0) = 0

(d) f is differentiable at (0,0)

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*Answer Key*

*Answer Key*

01. | (a), (b) | 06. | (b), (d) | 11. | (b), (c), (d) |

02. | (b) | 07. | (b), (d) | 12. | (b) |

03. | (a) | 08. | (a), (c), (d) | 13. | (a), (b), (c) |

04. | (a) | 09. | (b), (c), (d) | 14. | (a), (b), (d) |

05. | (c) | 10. | (a), (c), (d) | 15. | (c) |