Question

the graph of a function f(x) is given below. use the drop - down menus to select the correct answer. at point a, f(x) is positive. at point a, f(x) is choose your answer. at point b, f(x) is choose your answer. at point b, f(x) is choose your answer. at point c, f(x) is choose your answer. at point c, f(x) is choose your answer. at point d, f(x) is choose your answer. at point d, f(x) is choose your answer. at point e, f(x) is choose your answer. at point e, f(x) is choose your answer.

Explanation:

Step1: Recall function - value and derivative meaning

The value of $f(x)$ at a point is the $y$ - coordinate of the point on the graph. The sign of $f^{\prime}(x)$ is determined by the slope of the tangent line at the point.

Step2: Analyze point A

At point A, the $y$ - coordinate is above the $x$ - axis, so $f(x)$ is positive. The slope of the tangent line at point A is negative, so $f^{\prime}(x)$ is negative.

Step3: Analyze point B

At point B, the $y$ - coordinate is below the $x$ - axis, so $f(x)$ is negative. The slope of the tangent line at point B is negative, so $f^{\prime}(x)$ is negative.

Step4: Analyze point C

At point C, the $y$ - coordinate is below the $x$ - axis, so $f(x)$ is negative. The slope of the tangent line at point C is zero (horizontal tangent), so $f^{\prime}(x)$ is zero.

Step5: Analyze point D

At point D, the $y$ - coordinate is above the $x$ - axis, so $f(x)$ is positive. The slope of the tangent line at point D is zero (horizontal tangent), so $f^{\prime}(x)$ is zero.

Step6: Analyze point E

At point E, the $y$ - coordinate is above the $x$ - axis, so $f(x)$ is positive. The slope of the tangent line at point E is negative, so $f^{\prime}(x)$ is negative.

Answer:

At point A: $f(x)$ is Positive, $f^{\prime}(x)$ is Negative
At point B: $f(x)$ is Negative, $f^{\prime}(x)$ is Negative
At point C: $f(x)$ is Negative, $f^{\prime}(x)$ is Zero
At point D: $f(x)$ is Positive, $f^{\prime}(x)$ is Zero
At point E: $f(x)$ is Positive, $f^{\prime}(x)$ is Negative