   Chapter 3.1, Problem 43E

Chapter
Section
Textbook Problem

# A formula for the derivative of a function f is given. How many critical numbers does f have? f ′ ( x ) = 1 + 210 sin x x 2 − 6 x + 10

To determine

To find:

Critical numbers for f(x) by using the given  f'(x).

Explanation

1) Concept:

Find the values of x where f'x=0 and f'x doesn’t exist. That gives the critical numbers.

2) Definition:

A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist.

3) Given:

f'(x)=1+210sinxx2-6x+10

4) Calculation:

To find the critical numbers of f(x), find the values of x where f'x=0 and f'x doesn’t exists.

i) Check at which values of x the given f'x doesn’t exist.

f'x will be undefined when the denominator x2-6x+10 is 0. Since x2-6x+10  is not a factorable expression, here we don’t get values of x that makes the denominator zero.

Since we don’t get actual numbers, we need to use a discriminant to see if it has real zeros.

Discriminant = b2-4ac = -62-4*1*10= -4

Since its negative, it has only imaginary zeros because discriminant b2-4ac0, so the quadratic equation has imaginary zeros

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