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Find the maximum and minimum values of the function 9(0) = 50 6 sin(0) on the interval [o, 2Minimum valueMaximum value...

Question

Find the maximum and minimum values of the function 9(0) = 50 6 sin(0) on the interval [o, 2Minimum valueMaximum value

Find the maximum and minimum values of the function 9(0) = 50 6 sin(0) on the interval [o, 2 Minimum value Maximum value



Answers

Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints. $y=6 t-t^{2}, \quad[0,5]$

We have why? Equal to 60 minus T squared. And we're going to find the absolute maximum in absolute minimum In the interval from 4 to 6. It's possible that it could happen in a critical number. So let's take the derivative of the function Which would be 6 -2 t. And less determined where we get critical numbers. Now that could occur if the derivative is non differentiable, which 6 -2 T. is never non differentiable or where that derivative is equal to zero, we subtract the six from both sides, we get negative two. T equals negative six. And dividing by negative two gives us three. Now that gives us a quandary because three is not inside the interval. So since it's not inside the interval, it can't be an extremist for that interval. The only points we can check then are the endpoints of four and 6. If we fill in four we would get 24 -16 or eight. If we fill in six we get 36 -36 or zero, So the largest Y value is eight, so the absolute maximum occurs for eight, the smallest Y value is zero, So your absolute minimum occurs at 60.

Minimum. Oh, maximum value lies at vortex apartments. Are Texas a close to minus Willie? Right, Great toe. Nor that minimum occurs for quadratic function, which opens upward means bigger. Then deal maximum occurs for quadratic function, which opens downward. That implies a less than zero. I mean, why is it was toe six plus five x Esquire. You're the age was 25 greater than zero implies opens upward. That applies minimum occurs on the Y minimum. The minimum value will be at minus video Added weight to weigh. That means head a little. So why minimum medical to six plus Sybil deployed by zero square that people will you? Why, Min minimum is your question six.

We need to determine this function has a maximum or minimum value and then find it. So the number in front of X squared. That's R. A. Number. So if A. Is negative, our parabola opens downward and we have a maximum and it is positive. Our parabola opens upwards and it's a minimum in this case are A. Is negative. So we have a maximum. So to find the maximum value, we're going to do find our vertex by doing negative B over two a. And then evaluating the function at negative B over two. A. So we replace and we have- B. is six Over two times -1. Which is a. So this becomes three. So now we're going to evaluate the function F. At three. So this becomes negative three squared plus six times three plus two. So it becomes negative nine plus 18 plus two. So this becomes 9-plus 2 which is 11. So this is our maximum value.

We need to determine the following quadratic has a maximum or minimum. So when we look at this we have no A. X squared plus bx. So A. In this case is negative. Okay so when A. Is negative it opens down so are probably looks like this. So we're gonna have a maximum. So defiant. Our maximum point. We need to find our vertex which is negative B over two. A. And then we're going to evaluate the function at negative B over two. A. So we get negative B. Which is negative A negative 6/2 times A. Which is a negative one. We simplify that. We get six over negative two. So that's negative three. So now we take that negative three and plug it back into the function. So f at negative three is negative parentheses negative three squared minus six times negative three. So this becomes negative nine plus 18. Which is positive name. So that is our maximum point.


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