The function f(x) = 6x+9x^(-1) has 1 local max & 1 local min. Find the values?

Question

For some reason I keep getting stuck once I find the derivative. I have that f'(x)= 6-9x^(-2) but I can't find the local max and mins, no matter what I try .... thanks for your help!!!

Answer 1

Start with the first derivative :

f'(x) = 6-9/x^2

set f' = 0 = 6 - 9/x^2 ---> 6/9 =1/x^2 ---> 9/6 = x^2

x = +/- sqrt(3/2)

Now look at second derivative

f''(x) = 18/x^3

f''(x = -sqrt(3/2)) < 0 so the point x = -sqrt(3/2) is a maximum

f''(x = +sqrt(3/2)) >0 so the point x = +sqrt(3/2) is a minimum

Answer 2

f ' (x)= 6-9x^(-2) = 0 when
6 = 9/x^2 or
x^2 = 9/6, just cross-multiplty!

x = + - sqrt(9/6) or
x = + - 3/sqrt(6). These give the max and min,