Question
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.
List ALL intervals on which the functon f(x) is concave up and ALL intervals on which the function f(x) is concave down.
Answers
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
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.
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.
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