Questions: Question 10 Given: f''(x)=(x-5)(x-3)(x-1) a) f''(5)= b) f(5) is a Select an answer

Transcript text: Question 10 Given: $f^{\prime \prime}(x)=(x-5)(x-3)(x-1)$ a) $f^{\prime \prime}(5)=$ $\square$ b) $f(5)$ is a Select an answer

Solution

Solution Steps

To solve the given problem, we need to evaluate the second derivative of the function at a specific point and determine the nature of the function at that point.

a) To find $ f''(5) $, substitute $ x = 5 $ into the expression for $ f''(x) $.

b) To determine the nature of $ f(5) $, we need more information about the function $ f(x) $ itself, such as initial conditions or additional context, which is not provided in the question. Therefore, we can only evaluate $ f''(5) $ with the given information.

Step 1: Evaluate $ f''(5) $

Given the second derivative of the function:
\[ f''(x) = (x - 5)(x - 3)(x - 1) \]

Substitute $ x = 5 $ into the expression:
\[ f''(5) = (5 - 5)(5 - 3)(5 - 1) \]

Calculate:
\[ f''(5) = 0 \times 2 \times 4 = 0 \]

Step 2: Determine the Nature of $ f(5) $

Since $ f''(5) = 0 $, this indicates that the concavity of the function at $ x = 5 $ is neither concave up nor concave down. To determine the nature of $ f(5) $, we would typically need additional information such as the first derivative $ f'(x) $ or the function $ f(x) $ itself. However, with the given information, we can only conclude that the second derivative test is inconclusive at $ x = 5 $.