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University of California, Riverside

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Problem 41

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(x)=2 x-x^{2}, \quad-\infty<x \leq 2

$$

Answer

See the graph

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## Discussion

## Video Transcript

get some four problem 41. We work even the function FX, which is equal to two X minus X squared. So to answer the first part A In this problem, we need to take derivative of dysfunction, which is to minus two X and we let this derivative to be positive. Then we can find that X is smaller than one. So that means from next infinity to one, dysfunction will be increasing. And from 1 to 2, this function will be decreasing. And hence F one will be our loco extreme. Okay, Now, for part B need to identify whether the local stream here is is absolute the way to check. It's just Ah, um, check the maximum value off this whole entire function. Um, well, in this case, we there are two values that we need to check adversities to act one where that one is two minus one, which is one and the other. The other body we need a chap is the endpoint, which is to we'll have to be people plugging X equals two into our function. We will have zero. But dysfunction, as we know, is a parabola. So from so wouldn't x is equal to one. We reach our Mexico. And when we well, we checked the interval from 1 to 2, then it will be look like where it would be like this. So that means that one, it's the absolute maximum, which may say the except it is at his absolute.

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