PreCalculus
posted by Ann
Determine the derivative at the point (2,−43) on the curve given by f(x)=7−7x−9x^2.
I know that the answer is 43, but I was wondering if it was just a coincidence that the derivative at the point (2,−43) is 43, or is there a reason why 43 is the same as the yvalue of the original point.

Reiny
suppose you try another point, say (1, 9)
then
f ' (x) = 7  18x
f ' (1) = 7  9 = 16
Yup, just a coincidence.
Question: is there another point where this happens,
that is,
solve f(x) = f'(x)
7  7x  9x^2 = 7  18x
9x^2 + 11x +14 = 0
9x^2  11x  14 = 0
(x  2)(9x + 7) = 0
x = 2 or x = 7/9
we know about the x=2
when x = 7/9 , f(7/9) = 7  7(7/9)  9(49/81)
= 7
and f'(7/9) = 7  18(7/9) = 7
yup, two cases where it happens. 
Ann
this is weird...but thanks!

Eli
noooooooooo
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