Question

The graph of the function f(x)=(x-4)(x+1) is shown below. which statement about the function is true?

Answer 1

Answer:

(3) the function is decreasing for all real numbers of x where x< 1.5

Step-by-step explanation:

The function given in this problem is

$f(x)=(x-4)(x+1)$

which can be rewritten as:

$f(x)=x^2-4x+x-4=x^2-3x-4$

A function is:

- Increasing over a certain interval if the first derivative $f'(x)$ is positive in that interval

- Decreasing over a certain interval is the first derivative $f'(x)$ is negative in that interval

So we start by calculating the first derivative of this function. We get:

$f'(x)=2x^{2-1}-3x^{1-1}=2x-3$

This function is positive when:

$2x-3>0\\2x>3\\x>\frac{3}{2}$

Which means, when x > 1.5.

So the function is:

- Increasing for x > 1.5

- Decreasing for x < 1.5

So the correct option is

(3) the function is decreasing for all real numbers of x where x< 1.5