h(x)=x^2 +2x-9, g(-3)=2, g’(-3)=-6, f(x)=g(x)h(x) Find (dF)/(dX)|x=-39 F’(-3)= ????

Answer 1

h(x) = x^2 +2x-9

h(-3) = -6

g(-3)=2

g'(-3)=-6

f(x)=g(x)h(x)

F'(x)=f(x)

F'(-3) = f(-3) = g(-3)h(-3)

= 2((-3)^2 +2(-3) -9)

= 2(9-6-9)

= 2(-6)

=-12

Answer 2

Hi Joe!

h(x) = x² + 2x - 9

g(-3) = 2

g'(-3) = -6

f(x) = g(x)h(x)

I first tried g(x) = ax² + bx + c, but then you have too many variables.

g(x) = ax² + b works

g(-3) = a (-3)² + b

g(-3) = 9a + b

and since g(-3) = 2, 9a + b = 2

g'(x) = 2ax

g'(-3) = 2a(-3)

= -6a

and since g'(-3) = -6, -6a = -6 (a = 1)

Plug in for a in 9a + b = 2

9(1) + b = 2

9 + b = 2

b = -7

So g(x) = 1x² - 7

f(x) = g(x)h(x)

f(x) = (x²-7)(x²+2x-9)

f(x) = x⁴+2x³-9x²-7x²-14x+63

f(x) = x⁴+2x³-16x²-14x+63

df/dx = 4x³+6x²-32x-14

df/dx |x=-39 = 4(-39)³+6(-39)²-32(-39)-14 = -226,916

f'(-3) = 4(-3)³+6(-3)²-32(-3)-14 = 28

Hopefully didn't make any math errors!