This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Maxima and Minima of Two Variables – 1”.

1. Consider the f(x, y) = x^{2} + y^{2} – a. For what values of a do we have critical points for the function

a) independent of a

b) for any real number except zero

c) a ∊ (0, +∞)

d) a ∊ (-1, 1)

View Answer

Explanation: Consider

f

_{x}= 2x

and

f

_{y}= 2y

There is no a here. Thus, independent of a.

2. The critical point exist for the function f(x, y) = x^{n} + x^{n-1} y +……+y^{n} at (0,0)

a) True

b) False

View Answer

Explanation: Counter example is with n=1

f(x, y) = x + y.

3. f(x, y) = sin(x).cos(y) Which of the following is a critical point

a) (^{Π}⁄_{4}, ^{Π}⁄_{4})

b) (- ^{Π}⁄_{4}, ^{Π}⁄_{4})

c) (0, ^{Π}⁄_{2})

d) (0, 0)

View Answer

Explanation: f

_{x}= cos(x).cos(y) = 0

f

_{y}= – sin(x).sin(y)

→(x, y) = (0,

^{Π}⁄

_{2}).

4. The point (0,0) in the domain of f(x, y) = sin(xy) is a point of

a) Saddle

b) Minima

c) Maxima

d) Constant

View Answer

Explanation: Differentiating f

_{xx}= -y

^{2}.sin(xy)

f

_{yy}= -x

^{2}.sin(xy)

f

_{xy}= -yx.sin(xy)

Observe that f

_{xx}. f

_{yy}– (f

_{xy})

^{2}

Hence, it is a saddle point.

5. A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).

What is the slope of the line travelling on which would lead him to the lowest elevation.

a) There isn’t such a line

b) 1

c)-1

d) 0

View Answer

Explanation: Differentiating yields

f

_{xx}= -y

^{2}.sin(xy)

f

_{yy}= -x

^{2}.sin(xy)

f

_{xy}= -yx.sin(xy)

Observe that (0,0) is an inconclusive point

Hence, he will never reach the lowest elevation(because there isn’t such point.

6. let s(1) be the set of all critical points of f1(x, y) = g1(x).g2(y) and s(2) be the set of critical points of f2(g1(x), g2(y)) Which of the following is the right relation between s(1) and s(2), given that minimum number of elements in s(1) is 2.

a) s(1) = s(2)

b) s(1) ≠ s(2)

c) s(1) ∩ s(2) ≠ 0

d) depends on the functions

View Answer

Explanation: Differentiating f1(g1(x), g2(y)) with respect to x and y separately we get

d

_{x}= f1x g1x (x)

d

_{y}= f1y g1y (y)

This implies

g1x = 0

g1y = 0

Which are also the set of critical points of f1(x, y)

Thus we have the relation as s(1) ∩ s(2) ≠ 0.

7. f(x, y)=\(10y(10y-1)+(\frac{x^3sin(x^2)tan(x^3)}{(x-1)^3})-100y^2\). Find the critical points

a) (0,0)

b) (1,1)

c) (2,22)

d) None exist

View Answer

Explanation: Rewriting the function

f(x, y)=\((\frac{x^3sin(x^2)tan(x^3)}{(x-1)^3})-10y\)

Differentiating with respect to y we get

f

_{y}= -10

-10 ≠ 0

There exist no critical point.

8. Consider the vertical cone. The minimum value of the function in the region f(x,y) = c is

a) constant

b) 1

c) 0

d) -1

View Answer

Explanation: f(x,y) = c is a level curve over which the function has constant value

Hence, we have the answer as a constant.

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

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