The graph of the function [tex]f(x)=(x-4)(x+1)[/tex] is shown below.

Which statement about the function is true?
A. The function is increasing for all real values of [tex]x[/tex] where [tex]x\ \textless \ 0[/tex].
B. The function is increasing for all real values of [tex]x[/tex] where [tex]x\ \textless \ -1[/tex] and where [tex]x\ \textgreater \ 4[/tex].
C. The function is decreasing for all real values of [tex]x[/tex] where [tex]-1\ \textless \ x\ \textless \ 4[/tex].
D. The function is decreasing for all real values of [tex]x[/tex] where [tex]x\ \textless \ 1.5[/tex].

Find the derivative of the function: f ′ ( x ) = 2 x − 3 .
Determine the critical point by setting the derivative to zero: x = 1.5 .
Analyze the sign of the derivative in the intervals x < 1.5 (decreasing) and 1.5"> x > 1.5 (increasing).
Conclude that the function is decreasing for all real values of x where x < 1.5 , so the answer is T r u e ​ .

Explanation

Understanding the Problem We are given the function f ( x ) = ( x − 4 ) ( x + 1 ) and its graph. We need to determine which statement about the increasing or decreasing nature of the function is true. The options provide intervals where the function is claimed to be increasing or decreasing.

Finding the Derivative First, let's find the derivative of f ( x ) to determine where the function is increasing or decreasing. Expanding f ( x ) , we get f ( x ) = x 2 − 3 x − 4 . The derivative of f ( x ) is f ′ ( x ) = 2 x − 3 .

Finding Critical Points To find the critical points, we set f ′ ( x ) = 0 and solve for x : 2 x − 3 = 0 2 x = 3 x = 2 3 ​ = 1.5

Analyzing Intervals Now, we analyze the sign of f ′ ( x ) in the intervals defined by the critical point x = 1.5 .



For x < 1.5 , let's test x = 0 : f ′ ( 0 ) = 2 ( 0 ) − 3 = − 3 < 0 . So, f ( x ) is decreasing for x < 1.5 .
For 1.5"> x > 1.5 , let's test x = 2 : 0"> f ′ ( 2 ) = 2 ( 2 ) − 3 = 1 > 0 . So, f ( x ) is increasing for 1.5"> x > 1.5 .


Comparing with Statements Now, let's compare our findings with the given statements:


The function is increasing for all real values of x where x < 0 . This is false because we found that f ( x ) is decreasing for x < 1.5 .
The function is increasing for all real values of x where x < − 1 and where 4"> x > 4 . This is false because f ( x ) is increasing for 1.5"> x > 1.5 .
The function is decreasing for all real values of x where $-1

Answered by GinnyAnswer | 2025-07-04

The function f ( x ) = ( x − 4 ) ( x + 1 ) is decreasing for values of x between − 1 and 4 . Therefore, the correct statement about the function is option C. The function decreases in this interval until it reaches the critical point at 1.5 .
;

Answered by Anonymous | 2025-07-06