Question

question 9 of 10 view policies current attempt in progress at exactly two of the labeled points in the figure below, the derivative $f$ is 0; the second derivative $f$ is not zero at any of the labeled points. give the signs of $f$, $f$, $f$ at each marked point. etextbook and media save for later using multiple attempts will impact your score. 5% score reduction after attempt 5 attempts: 0 of 15 used submit answer

Explanation:

Step1: Analyze point A

The function value \(f(A)\) is below the \(x -\)axis, so \(f(A)<0\). The slope of the tangent line at \(A\) is positive, so \(f'(A)>0\). The curve is concave - down at \(A\), so \(f''(A)<0\).

Step2: Analyze point B

The function value \(f(B)\) is above the \(x -\)axis, so \(f(B)>0\). The slope of the tangent line at \(B\) is \(0\) (local maximum), so \(f'(B) = 0\). The curve is concave - down at \(B\), so \(f''(B)<0\).

Step3: Analyze point C

The function value \(f(C)\) is above the \(x -\)axis, so \(f(C)>0\). The slope of the tangent line at \(C\) is negative, so \(f'(C)<0\). The curve is concave - down at \(C\), so \(f''(C)<0\).

Step4: Analyze point D

The function value \(f(D)\) is below the \(x -\)axis, so \(f(D)<0\). The slope of the tangent line at \(D\) is \(0\) (local minimum), so \(f'(D)=0\). The curve is concave - up at \(D\), so \(f''(D)>0\).

Answer:

Point	\(f\)	\(f'\)	\(f''\)
\(B\)	Positive	Zero	Negative
\(C\)	Positive	Negative	Negative
\(D\)	Negative	Zero	Positive