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
[tex]f(x)=(x-4)(x+1)[/tex]
which can be rewritten as:
[tex]f(x)=x^2-4x+x-4=x^2-3x-4[/tex]
A function is:
- Increasing over a certain interval if the first derivative [tex]f'(x)[/tex] is positive in that interval
- Decreasing over a certain interval is the first derivative [tex]f'(x)[/tex] is negative in that interval
So we start by calculating the first derivative of this function. We get:
[tex]f'(x)=2x^{2-1}-3x^{1-1}=2x-3[/tex]
This function is positive when:
[tex]2x-3>0\\2x>3\\x>\frac{3}{2}[/tex]
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
