Questions: The twice-differentiable function f is shown below on the domain (-9,9). The function f has points of inflection at x=-7.6, x=-2.4, x=2, x=6, shown with small green circles on the graph. Determine what could be said about the values of f(6), f'(6), and f''(6).

The twice-differentiable function f is shown below on the domain (-9,9). The function f has points of inflection at x=-7.6, x=-2.4, x=2, x=6, shown with small green circles on the graph. Determine what could be said about the values of f(6), f'(6), and f''(6).

Transcript text: The twice-differentiable function $f$ is shown below on the domain $(-9,9)$. The function $f$ has points of inflection at $x=-7.6, x=-2.4, x=2, x=6$, shown with small green circles on the graph. Determine what could be said about the values of $f(6), f^{\prime}(6)$, and $f^{\prime \prime}(6)$.

Solution

Solution Steps

Step 1: Find f(6)

The graph passes through the point (6, -5). Thus, f(6) = -5.

Step 2: Find f'(6)

The graph is decreasing at x = 6, meaning the slope of the tangent line at x=6 is negative. Thus, f'(6) < 0.

Step 3: Find f''(6)

The function has an inflection point at x = 6. This means the concavity of the graph changes at this point. Looking at the graph, the concavity changes from concave up (f''(x) > 0) to concave down (f''(x) < 0) at x = 6. Thus, f''(6) = 0.

Final Answer

f(6) = -5, f'(6) < 0, and f''(6) = 0.

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