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Problem 71

Determine the values of constants $a$ and $b$ so that $f(x)=a x^{2}+b x$

has an absolute maximum at the point $(1,2) .$

Answer

$a=-2$ and $b=4$

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## Discussion

## Video Transcript

in order for function to have a maximum at a given 0.2 conditions must be met. The first that the first derivative, evaluated at a given point, is equal to zero the second that the second new evidence evaluated at the given point is less than zero. So if we look at a given function at the 0.1 comma too, your first see but two given that is the why is equal to a Times X, which is one squared plus b times X, just one. And now if we look at the first derivative a crime of X but evaluated at X equals one, we see that it is equal to two a times X, which is one plus b. So if we want to solve these two equations, we see that too is equal to a plus B. We also know because this isn't maximum zero is equal to two a plus B. So if we solve these equations, we see that B is equal to negative to a. Thus, two is equal to a plus. Negative to a and A is equal to negative two. If a is equal to negative too. If we look at this first equation we see that therefore be is equal to four given a which gives us values for A and B and just toe double. Check these answers. If we look at the second derivative, a double prime of X, which is equal to two A well to A is equal to two times negative two, which is equal to negative full, which is indeed less than zero. And thus we see that the values for A and B negative two and four respective.

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