the graph below shows the derivative f (x) of the function f (x). at which of the following x values does…
the graph below shows the derivative f (x) of the function f (x). at which of the following x values does the original function f (x) have a relative maximum? x=1 x=4.3 x=0.3 x=3.3 x=1.8
Answer
Explanation:
Step1: Recall the first - derivative test
The first - derivative test states that if (f^{\prime}(x)) changes sign from positive to negative at a point (x = c), then (f(x)) has a relative maximum at (x = c).
Step2: Analyze the sign of (f^{\prime}(x)) around each option
- For (x = 1): (f^{\prime}(x)) changes sign from positive to negative at (x = 1). Before (x = 1), (f^{\prime}(x)>0) (the graph of (f^{\prime}(x)) is above the (x) - axis), and after (x = 1), (f^{\prime}(x)<0) (the graph of (f^{\prime}(x)) is below the (x) - axis).
- For (x = 4.3): (f^{\prime}(x)) is positive and increasing, so there is no relative maximum.
- For (x = 0.3): (f^{\prime}(x)) is positive and increasing, so there is no relative maximum.
- For (x = 3.3): (f^{\prime}(x)) is negative and increasing, so there is no relative maximum.
- For (x = 1.8): (f^{\prime}(x)) is negative, so there is no relative maximum.
Answer:
(x = 1)