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QUESTION: 1

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the radius of the semicircular opening of the window to admit maximum light.

Solution:

QUESTION: 2

The point of local maxima for the function f(x) = sinx. cos x is

Solution:

f(x) = sinx.cosx

f’(x) = -sin^{2}x + cos^{2}x

-sin^{2}x + cos^{2}x = 0

sin^{2}x = cos^{2}x

tan^{2}x = 1

tanx = +-1

for tan x = 1, x = π/4

for tan x = -1, x = 3π/4

At x = π/4, f(π/4) = ½

At x = 3π/4, f(3π/4) = -½

At π/4 is the local maxima.

QUESTION: 3

A real number x when added to its reciprocal give minimum value to the sum when x is

Solution:

QUESTION: 4

The sum of two positive numbers is 20. Find the numbers if their product is maximum

Solution:

QUESTION: 5

The function f(x) = log x

Solution:

f(x) = log x

f’(x) = 1/x

Putting f’(x) = 0

1/x = 0

x = ∞ (this is not defined for x)

So, f(x) does not have minima or maxima

QUESTION: 6

f(x) = x^{5} – 5x^{4} + 5x^{3} – 1. The local maxima of the function f(x) is at x =

Solution:

f(x) = x^{5} - 5x^{4} + 5x^{3} - 1

On differentiating w.r.t.x, we get

f'(x) = 5x^{4} - 20x^{3} + 15x^{2}

For maxima or minima,f'(x) = 0

⇒ 5x^{4} - 20x^{3} + 15x^{2} = 0

⇒ 5x^{2}(x^{2}-4x+3) = 0

⇒ 5x^{2}(x-1)(x-3) = 0

⇒ x = 0,1,3

So, y has maxima at x = 1 and minima at x = 3

At x = 0, y has neither maxima nor minima, which is the point of inflection .

f(1) = 1-5+5-1=0,which is local maximum value at x = 1.

QUESTION: 7

How many units should be sold so that a company can make maximum profit if the profit function for x units is given by p(x) = 25 + 64x - x^{2}

Solution:

QUESTION: 8

Find two numbers whose sum is 24 and product is a large as possible.

Solution:

QUESTION: 9

A point c in the domain of a function f is called a critical point of f if

Solution:

A point C in the domain of a function f at which either f(c) = 0 or f is not differentiable.

The point f is called the critical point.

c is called the point of local maxima

If f ′(x) changes sign from positive to negative as x increases through c, that is, if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c.

c is called the point of local minima

If f ′(x) changes sign from negative to positive as x increases through c, that is, if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c.

c is called the point of inflexion

If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima.

QUESTION: 10

Find two positive numbers x and y such that x + y = 60 and xy^{3} is maximum

Solution:

two positive numbers x and y are such that x + y = 60.

x + y = 60

⇒ x = 60 – y ...(1)

Let P = xy^{3}

∴ P =(60 – y)y^{3} = 60y^{3} – y4

Differentiating both sides with respect to y, we get

For maximum or minimum dP/dy = 0

⇒ 180y^{2} - 4y^{3} = 0

⇒ 4y^{2} (45 - y) = 0

⇒ y = 0 or 45 - y = 0

⇒ y = 0 or y = 45

⇒ y = 45 (∵ y = 0 is not possible)

Thus, the two positive numbers are 15 and 45.

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