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QUESTION: 1

Consider the function f(x) = x^{2} – x – 2. The maximum value of f(x) in the closed interval [–4, 4] is

Solution:

∴ f (x )has minimum at x= 1 / 2 It Shows that a maximum value that will be at x = 4 or x = - 4

At x = 4, f (x )= 10

∴ At x= −4, f (x ) = 18

∴ At x= −4, f (x ) has a maximum.

QUESTION: 2

Solution:

QUESTION: 3

Solution:

QUESTION: 4

The function f(x) = x^{3}- 6x^{2}+ 9x+25 has

Solution:

QUESTION: 5

The function f(x,y) = 2x^{2} +2xy – y^{3} has

Solution:

QUESTION: 6

Equation of the line normal to function f(x) = (x-8)^{2/3}+1 at P(0,5) is

Solution:

QUESTION: 7

The distance between the origin and the point nearest to it on the surface z^{2} = 1 + xy is

Solution:

or pr – q^{2} = 4 – 1 = 3 > 0 and r = +ve

so f(xy) is minimum at (0,0)

Hence, minimum value of d^{2} at (0,0)

d2 = x^{2} + y^{2} + xy + 1 = (0)^{2} + (0)^{2} + (0)(0) + 1 = 1

Then the nearest point is

z^{2} = 1 + xy = 1+ (0)(0) = 1

or z = 1

QUESTION: 8

Given a function

The optimal value of f(x, y)

Solution:

QUESTION: 9

For the function f(x) = x^{2}e-x, the maximum occurs when x is equal to

Solution:

QUESTION: 10

A cubic polynomial with real coefficients

Solution:

So maximum two extrema and three zero crossing

QUESTION: 11

Given y = x^{2} + 2x + 10, the value of

Solution:

QUESTION: 12

Consider the function y = x^{2} – 6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 is

Solution:

QUESTION: 13

^{2} – 6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 is

Solution:

**Correct Answer :- b**

QUESTION: 14

Solution:

QUESTION: 15

The magnitude of the gradient of the function f = xyz^{3} at (1,0,2) is

Solution:

QUESTION: 16

The expression curl (grad f), where f is a scalar function, is

Solution:

QUESTION: 17

If the velocity vector in a two – dimensional flow field is given by the vorticity vector, curl

Solution:

QUESTION: 18

The vector field

Solution:

QUESTION: 19

The angle between two unit-magnitude co-planar vectors P (0.866, 0.500, 0) and Q (0.259, 0.966, 0) will be

Solution:

QUESTION: 20

Stokes theorem connects

Solution:

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