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$$y=-x^2+2px+q$$ has a maximum value of 5 when x = 3.

Find the values of p and q

First of all:

dy/dx=-2x+2p

We know that dy/dx=0 for x=3 as this means the gradient is 0 (turning point)

So:

0=-2 * 3 + 2p

0=-6+2p

2p=6

p=3

Now we know the graph goes through the point (3,5) and p=3 so the equation becomes:

5 = -(3^2) + 2 * 3 * 3 + q

5=-9+18+q

q=-4

You could also do this by using x=-b/2a where x is the x value of the turning point

So: 3=-2p/-2

p=3

Use the same method as before to find q
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