Question

use the graph below to fill in each entry of the grid with positive, negative or zero.

Explanation:

Step1: Analyze \(f\) at \(A\), \(B\), \(C\)

• At \(A\): The point \(A\) is below the \(x -\)axis. So \(f(A)<0\) (negative).
• At \(B\): The point \(B\) is above the \(x -\)axis. So \(f(B)>0\) (positive).
• At \(C\): The point \(C\) is above the \(x -\)axis. So \(f(C)>0\) (positive).

Step2: Analyze \(f'\) at \(A\), \(B\), \(C\)

• The first - derivative \(f^{\prime}(x)\) represents the slope of the tangent line to \(y = f(x)\).
• At \(A\): The function \(y = f(x)\) has a horizontal tangent at a local minimum. So \(f^{\prime}(A)=0\).
• At \(B\): The function \(y = f(x)\) is increasing (the slope of the tangent line is positive). So \(f^{\prime}(B)>0\) (positive).
• At \(C\): The function \(y = f(x)\) has a horizontal tangent at a local maximum. So \(f^{\prime}(C)=0\).

Step3: Analyze \(f''\) at \(A\), \(B\), \(C\)

• The second - derivative \(f^{\prime\prime}(x)\) represents the concavity of \(y = f(x)\). If the function is concave up, \(f^{\prime\prime}(x)>0\); if concave down, \(f^{\prime\prime}(x)<0\).
• At \(A\): The function \(y = f(x)\) is concave up (shaped like a "cup"). So \(f^{\prime\prime}(A)>0\) (positive).
• At \(B\): The function \(y = f(x)\) is concave down (shaped like a "cap"). So \(f^{\prime\prime}(B)<0\) (negative).
• At \(C\): The function \(y = f(x)\) is concave down (shaped like a "cap"). So \(f^{\prime\prime}(C)<0\) (negative).

Answer:

	\(f\)	\(f^{\prime}\)	\(f^{\prime\prime}\)
\(B\)	positive	positive	negative
\(C\)	positive	zero	negative