This set of Engineering Mathematics Questions and Answers for Entrance exams focuses on “Maxima and Minima of Two Variables – 2”.

1. Find the critical points of the function

a) (0,0)

b) (0,-90)

c) (90, 0)

d) None exist

View Answer

Explanation:Find f

_{x}= 10

10 ≠ 0

Hence, no points exist.

2. For function f(x, y) = sin^{-1}(x^{2} + y^{2}) critical points are found. Now a new graph g(x, y) is formed by coupling graphs f(x, y) and f(x, y) = – sin^{-1}(x^{2} + y^{2}). What are the critical points of g(x, y)

a) (0,0)

b) There are infinite such points

c) Only positive (x, y) are critical points

d) (90,-90)

View Answer

Explanation: The function takes constant values along a circle(observe the function)

But it is composed of arc sine function. Hence, we will have critical points at equal intervals

Hence, there are infinite such points.

3. Consider the circular region x^{2} + y^{2} = 81, What is the maximum value of the function

f(x, y) = x^{6} + y^{2}(3x^{4} + 1) + x^{2}.(3y^{4} + 1) + y^{6}

a) 90

b) 80

c) 81 + 81^{3}

d) 100

View Answer

Explanation: Rewrite the function as

f(x, y) = x

^{2}+ y

^{2}+ (x

^{2}+ y

^{2})

^{3}

Put x

^{2}+ y

^{2}= 81

= 81 + 81

^{3}.

4. What is the maximum value of the function

f(x, y) = x^{2}(1 + 3y) + x^{3} + y^{3} + y^{2}(1 + 3x) + 2xy over the region x=0; y=0; x + y=1

a) 0

b) -1

c) Has no maximum value

d) 2

View Answer

Explanation:

Rewrite the function as

f(x, y) = (x + y)

^{2}+ (x + y)

^{3}

Put x + y = 1

= 2.

5. If the Hessian matrix of a function is zero then the critical point is

a) It cannot be concluded

b) Always at Origin

c) Depends on Function

d) (100,100)

View Answer

Explanation: If the Hessian matrix is zero then the second derivative test fails and nothing can be said about the crtical points.

6. The maximum value of the function

f(x, y) = sin(x).cos(2y).cos(x + 2y) + sin(2y).cos(x + 2y).cos(x) in the region x=0; y=0; x+2y = 3

a) 90

b) cos(1)

c) sin(1).cos(1)

d) sin(3).cos(3)

View Answer

Explanation:Rewrite the function as

f(x, y) = cos(x + 2y) * (sin(x).cos(2y) + cos(x).sin(2y))

f(x, y) = cos(x + 2y).sin(x + 2y)

Put x+2y = 3

= sin(3).cos(3).

7. Find the minimum value of the function f(x, y) = x^{2} + y^{2} +199 over the real domain

a) 12

b) 13

c) 0

d) 199

View Answer

Explanation: Find

f

_{x}= 2x

f

_{y}= 2y

The critical point is

x=0

y=0

(0,0) is the critical point

Put it back into the function we get

z = 0 + 0 + 199 = 199 is the required minimum value.

8. What is the maximum value of the function f(x, y) = 3xy + 4x^{2}y^{2} in the region

x=0; y=0; 2x + y = 2

a) 1

b) 0

c) 100

d) 10

View Answer

Explanation: Differentiating we have

f

_{x}= 3y + 8xy

^{2}= 0

f

_{y}= 3x + 8x

^{2}y

x = 0

y = 0

(0,0) lies in the region

Substitute x=0

f(0, y) = 0

Substitute y=0

f(x, 0) = 0

Substitute y = 2 – 2x

is the maximum value

**Sanfoundry Global Education & Learning Series – Engineering Mathematics.**

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