Question

Find the relative maximum and minimum value if the following one variable function (If there is any):

a) y = x^2 + 2x + 1

b) y = 6x^2 - 12x

c) y = x^2 + 2x + 1

d) y = x^4

Answer 1

a) y is minimum when x = -1 and minimum value of y = 0

Equating to 0, we get,

Hence x= -1 is a minimum

When x= -1,

b) y is minimum when x = 1 and minimum value of y = -6

Equating to 0, we get,

Hence x= 1 is a minimum

When x=1,

c) y is minimum when x = -1 and minimum value of y = 0

Equating to 0, we get,

Hence x= -1 is a minimum

When x = -1,

d) y is minimum at x=0 and the minimum value of y =0

Equating to 0, we get,

The second derivative of the function is 0 at x=0. Therefore, we cannot be sure if x=0 is a minimum or maximum or a point of inflection. First, we will check if x=0 is an inflection point. For this, we need to see if second derivative is of same sign on both sides of x=0.

At x= -1,

At x= 1,

The second derivative do not change sign on both sides of x=0. Hence we can conclude x=0 is not an inflection point. Now, let us see if x=0 is a minimum or maximum. For that we need to calculate the first derivative on both sides of x=0.

At x= -1,

At x= 1,

That is slope changes from negative to positive at x=0. hence this point is a minimum.

At x=0,