a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

$$

f(\theta)=3 \theta^{2}-4 \theta^{3}

$$

$$

=\frac{1}{2}

$$

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Okay, so in problem 23 we have the function at feta, which is equal to three D in a square, minus four. You like you? Okay, Now for part A, we take that The river tip off this function. We have 60 there minus 12 Peter Square. And we let this to be positive. So to solve these inequality, we take out six feta. Now that it will be one minus two feta. Okay, now, again, this is Parappa with solution Zero and 1/2 Should be looked like this. Sorry, I would draw it again. OK, so that's our Perella. So that means, um with you guys in 0 to 1 have dysfunction will be increasing because Perella from 0 to 1/2 we'll have the positive output. Now would he dies from naked ability to zero union want have to ability. Dysfunction will be decreasing now, correspondingly, our political values or our extreme values. Locally, scream bells will be f zero and have 1/2. But we don't know yet whether these two red bettors are absolute batteries. So we can to see that we can draw the graph off this responding function. Um, by with those information from zero to from negative protective initative zero dysfunction will be decreasing and from from zero to say this is zero and from 0 to 1 have we have increasing function said this one have now from one have to affinity. We have we have ah increasing decreasing function. So this approximately our our grandpa function. So that means we observed a graph. We can conclude that there's no absolute maximum and absolute minimum, so no absolute extreme.

University of California, Riverside