Suppose you know that f(x) is an odd functon on the domain of all real numbers and that the function is concave up on the intervals 0 < x < 3 and 5 < x and concave down on the interval 3 < x < 5.
List ALL intervals on which the functon f(x) is concave up and ALL intervals on which the function f(x) is concave down.
Nevermind, I figured it out. Since the function is odd it must be symmetric about the origin, so:
concave up: -5 < x < -3 and 0 < x < 3 and x > 5
concave down: x < -5 and -3 < x < 0 and 3 < x < 5
We don't really know anything about f(x) for x>5, but let's assume it stays concave down.
Since f is odd, it is radially symmetric about the origin.
So, f is concave down on -3 < x < 0
and f is concave up on -5 < x < -3
We will assume it stays concave up for x < -5.
Well, well, well, we have a function that's trying to be all fancy with its concavity! Let's break it down, shall we?
First, let's focus on the intervals where our function is concave up. We know that it's concave up on the interval 0 < x < 3, and also on the interval 5 < x. So, our function is feeling happy and smiling on these intervals.
Now, when it comes to the intervals where the function is concave down, we know that it's concave down on the interval 3 < x < 5. It's like our function ate a sour clown nose or something. Sad clown vibes, I tell ya!
To sum it up:
Concave up intervals: 0 < x < 3 and 5 < x
Concave down interval: 3 < x < 5
Hope that clears it up for you in a fun and whimsical way!
To determine the intervals on which the function f(x) is concave up or concave down, we need to analyze the second derivative of the function.
1. Start by finding the first derivative of f(x), denoted as f'(x). Since f(x) is an odd function, we know that f'(x) will be an even function.
2. Next, find the second derivative of f(x), denoted as f''(x), by taking the derivative of f'(x). Since f'(x) is an even function, f''(x) will be an odd function.
3. Analyze the sign changes of f''(x) to determine the intervals of concavity.
- When f''(x) is positive, the function f(x) is concave up.
- When f''(x) is negative, the function f(x) is concave down.
4. Identify the intervals on which f''(x) changes sign, which will indicate the intervals of concavity.
Given that f(x) is concave up on the intervals 0 < x < 3 and 5 < x, and concave down on the interval 3 < x < 5, we can conclude the following:
- The function f(x) is concave up on the interval 0 < x < 3 and 5 < x.
- The function f(x) is concave down on the interval 3 < x < 5.