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.

$$

H(t)=\frac{3}{2} t^{4}-t^{6}

$$

See the graph

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Okay, so for problem 29 the function given to us it's ht divine to be 3/2 t to the power of four minus T to six. Get now for part a Again. We need to take the directive of HT. So in this case, it will be six t cute minus six t 2 to 5. Okay. Now, to solve these inequality, we let us to be positive. Not itself is inequality would take out 60. Cute first. Now we have one plus tee times. One might have stayed inside. Okay, Now again, we need to have two conditions to consider. Now, the first condition is all positive. That is, That is to say all of these three expressions are positive. Now, first t t the cute will be a positive. That means tea should be positive. And one plus 10 should be positive and one minus teacher to be positive and combining these three, we have t from 0 to 1. This function will be always we always be catapulted. So this is our solution to the first condition. Now the second condition that is to say we have two off. Three are negative. Okay, So what What we need to do is to first assume the 1st 2 expressions are negative. That is T is negative and one trustee is negative. And the last 11 modesty has to be positive and the wine key And then the second set of possibilities. The second set of possibility is tease connective and one plus t It is positive. And on minus C is negative. The last possibility which is t loss too. One last ti not to and on minus t negative. So that gives our, um that gives our three possibilities. We check each one of them. The first possibility will give us tow uh, t smarter than makes one. The second set up inequalities that he was as nothing The same s the third set up inequalities. So combine he's combine these two results Now we can conclude when t is from next to infinity to to a negative one you and you 0 to 1. Then this function will be increasing. Otherwise, from 1 to 0 you again one to infinity dysfunction will be decreasing Now The next thing is to your party we need to find a local extreme. So just by observing these intervals, we can find locus locally screams. There are three Asia connective one and a zero and each one. And to find the the absolute stream, we can just simply draw the graph of dysfunction that is, we have three low three local extreme making 10 and one from naked wanting from next infinity to next one, we have increasing function and then we have a decreasing function and then we have an increasing function and then he is decreasing. So absolutely there. There's no there's no no absolute minimum. But we do have a have a next minute. We do have a next with absolute maximum way. Just need to compare, uh, have a snack of one h one. Which one is bigger. So it's a connected one re plugging to our our original function. We will get 1/2 now each one. We also be 1/2. So that means these two are all absolute values. Absolute extremes. So that means that these two are both absolute extremes

University of California, Riverside