Find g'(5)?

Question

If f(x)=x^3-3x^2+8x+5 and g(x)=f^-1 (x), find g'(5).

 

Answer: 1/8

 

Please show all your work.

Answer 1

f(x) = x^3 – 3x^2 + 8x + 5 .

g(x) is inverse of f(x).

Find g’(5).

When does f(x) = 5?

5 = x^3 – 3x^2 + 8x + 5

0 = x(x^2 – 3x + 8)
(-3)^2 – 4(1)(8) = 9 – 32 < 0
So the only real x that makes f = 5 is 0; i.e.,
f(0) = 5

The corresponding point on g(x) is g(5) = 0.

The slopes of corresponding points on inverses are reciprocals, so

g’(5) = 1/f’(0)

f’(x) = 3x^2 – 6x + 8

f’(0) = 8

g’(5) = 1/8

 

Here's a GeoGebra sketch:

http://www.wyzant.com/resources/files/264583/derivative_of_inverse

 

f(x) is in blue, f'(x) red, g(x) green.

 

I should have hidden point D; disregard it, it's only an aide to construction.