help-critical points-extrema

Question

Answer 1

(1)

Step 1:

The function is .

Critical number :

A critical number of a function f  is a number c in the domain of f  such that either or does not exist.

is differentiable at all values of x because it is a polynomial.

Therefore the solutions of are the critical numbers.

Differentiate on each side with respect to x.

Step 2:

.

Equate to zero.

and

and

Critical numbers are and .

Solution :

Critical numbers are and .

Answer 2

(2)

Step 1:

The function is .

Critical number :

A critical number of a function f is a number c in the domain of f such that either or does not exist.

Therefore the solutions of are the critical numbers.

Differentiate on each side with respect to t.

Step 2:

.

Equate to zero.

and

and

Critical numbers are and .

Solution :

Critical numbers are and .

Answer 3

(3)

Step 1:

The function is .

Critical number :

A critical number of a function f  is a number c  in the domain of f  such that either or does not exist.

is differentiable at all values of x because it is a polynomial.

Therefore the solutions of are the critical numbers.

Differentiate on each side with respect to a.

Step 2:

.

Equate to zero.

The general solution for sine function, if  then .

Critical number is .

Solution :

Critical number is .

Answer 4

(4)

Step 1:

The function is .

Critical number :

A critical number of a function f  is a number c in the domain of f  such that either or does not exist.

is differentiable at all values of x because it is a polynomial.

Therefore the solutions of are the critical numbers.

Differentiate on each side with respect to x.

Step 2:

.

Equate to zero.

Critical number is .

Solution :

Critical number is .