Questions: If a function f is increasing on (a, b) and decreasing on (b, c), then what can be said about the local extremum of f on (a, c) ? The local maximum of f on (a, c) is f(b). The local minimum of f on (a, c) is f(b). The local maximum of f on (a, c) is f(c). The local maximum of f on (a, c) is f(a). The local minimum of f on (a, c) is f(c).

If a function f is increasing on (a, b) and decreasing on (b, c), then what can be said about the local extremum of f on (a, c) ?
The local maximum of f on (a, c) is f(b).
The local minimum of f on (a, c) is f(b).
The local maximum of f on (a, c) is f(c).
The local maximum of f on (a, c) is f(a).
The local minimum of f on (a, c) is f(c).

Transcript text: If a function $f$ is increasing on $(a, b)$ and decreasing on $(b, c)$, then what can be said about the local extremum of $f$ on $(a, c)$ ? The local maximum of $f$ on $(a, c)$ is $f(b)$. The local minimum of $f$ on $(a, c)$ is $f(b)$. The local maximum of $f$ on $(a, c)$ is $f(c)$. The local maximum of $f$ on $(a, c)$ is $f(a)$. The local minimum of $f$ on $(a, c)$ is $f(c)$.

Solution

Solution Steps

To determine the local extremum of the function $ f $ on the interval $ (a, c) $, we need to analyze the behavior of the function. Since $ f $ is increasing on $ (a, b) $ and decreasing on $ (b, c) $, the function reaches a local maximum at $ b $. Therefore, $ f(b) $ is the local maximum on $ (a, c) $.

Step 1: Analyze the Function Behavior

The function $ f $ is given to be increasing on the interval $ (a, b) $ and decreasing on the interval $ (b, c) $. This implies that the function reaches a peak at $ x = b $, where it transitions from increasing to decreasing.

Step 2: Determine the Local Maximum

Since the function is increasing up to $ x = b $ and then decreasing after $ x = b $, the value $ f(b) $ is a local maximum on the interval $ (a, c) $.

Step 3: Evaluate the Function at $ b $

Given the function $ f(x) = -1 \cdot (x - 2)^2 + 4 $, we evaluate it at $ x = b = 2 $: \[ f(2) = -1 \cdot (2 - 2)^2 + 4 = 4 \]