Determine the derivative at the point (2,−43) on the curve given by f(x)=7−7x−9x^2.

Question

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 y-value of the original point.

Answer 1

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.

Answer 2

this is weird...but thanks!

Answer 3

noooooooooo