Question

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

Explanation:

Step1: Analyze \( f \)

• At point \( A \): The \( y - \)coordinate (\( f(A)\)) is below the \( x - \)axis, so \( f(A)<0\) (negative).
• At point \( B \): The \( y - \)coordinate (\( f(B)\)) is on the \( x - \)axis, so \( f(B) = 0\).
• At point \( C \): The \( y - \)coordinate (\( f(C)\)) is above the \( x - \)axis, so \( f(C)>0\) (positive).

Step2: Analyze \( f' \)

• The first - derivative \( f'(x)\) represents the slope of the tangent line.
• At point \( A \): The function is increasing (slope of the tangent line is positive), so \( f'(A)>0\) (positive).
• At point \( B \): The function has a local minimum. The slope of the tangent line at a local minimum is \( 0\), so \( f'(B)=0\).
• At point \( C \): The function is decreasing (slope of the tangent line is negative), so \( f'(C)<0\) (negative).

Step3: Analyze \( f'' \)

• The second - derivative \( f''(x)\) represents the concavity. If \( f''(x)>0\), the function is concave up; if \( f''(x)<0\), the function is concave down.
• At point \( A \): The function is concave up (like a cup), so \( f''(A)>0\) (positive).
• At point \( B \): The function is concave up (since it is a local minimum of a smooth function, and the concavity is the same in the neighborhood of a local minimum for a twice - differentiable function), so \( f''(B)>0\) (positive).
• At point \( C \): The function is concave down (like a cap), so \( f''(C)<0\) (negative).

Answer:

	\( f \)	\( f' \)	\( f'' \)
\( B \)	zero	zero	positive
\( C \)	positive	negative	negative