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Problem 72

Determine the values of constants $a, b, c,$ and $d$ so that
$f(x)=a x^{3}+b x^{2}+c x+d$ has a local maximum at the point
$(0,0)$ and a local minimum at the point $(1,-1)$ .

$a=-\frac{2}{5}, b=-\frac{3}{5}$ and $c=d=0$

## Discussion

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## Video Transcript

Okay, So you want to find the values of a be sandy? So that path of X, she's just a general cubic X squared CX plus the They want a local max zero zero and a local men. That's it. One next one. Okay, this is going to be crazy. All right, So what can we do? Well, let's just pretend like we're finding the local district. So let's take the derivative. And then we're gonna have three a x squared plus two b x plus sing, okay. And so cool. Because we have local max at dear zero and a local man. Hat one negative one zero and one should be critical points. So f crime of zero, which is just see chutikul zero. And then f crime of one, which is three a plus to be should equal zero as well. So we see that's CIA zero. That's nice. That's one fourth of the way done. And then we see we have this equation. Okay, But now we also won the values at zero to be zero and at one to be minus one. So what causes that? Death of zero, which is D is equal to zero So that tells us sixty zero and then if we pull it in one so half of one. And now remember, we are You know that scene two year zero? That's just gonna be a plus B. That should be zero. All right. Excellent. So now we're going to solve for angry. Hey, so this is probably easier. So you see from here, that is minus B. So if I played in minus B for a lot of negative three B plus two B equals zero, I hope you know what? I'm instinct. People think I'm crazy. Now. A Busby equals negative. One breath of one needs to be negative one. All these other things are going to be zero. But effort, one needs to be minus one. So to get tape us B with my one. Here we go. So again, from here we get a is minus one minus being. Now, let me plug that in for a saw for being Kitt, we just move all this over here. So I get to be is three plus b over. Yes, three of me. Corby is negative. Three. Just strike this over. Get being the subject. Three Get negative three. And so therefore, a is minus one minus. Thank you. Three negative. One plus three witches too. So it looks like Ah, ffx should be two x cubed, minus three X squared. Okay, Now, if we want, we can check our answer. So if we take the derivative, we get f prime of X is six x squared, minus six X when there's a crime equal to zero. Well, the compactor office xx next minus one. So that's what X is your and X is one notice that this is ah, positive. Cubic. So, you know, we sort of know the behavior. Offhand, you're the one we know. It's going to be increasing the decreasing the increasing. So we have a local max at zero, and then we just plug in zero, get zero, can there? We want a local men. Well, okay, so we know there's a local men at one, cause we're decreasing than increasing. What's the value? Their will to minus three negative one. So everything checks out