First, differentiate. Let f(x) = (g(x))/(h(x)). Then g(x) = x^3 and h(x) = sqrt(x + 25).
We can differentiate g(x) using the power rule. g'(x) = 3x^2
We need the chain rule for h(x). Let y = sqrt(u) = u^(1/2) and u = x + 25. Then dy/(du) = 1/2u^(-1/2) = 1/(2u^(1/2). and (du)/dx = 1.
h'(x) = 1/(2u^(1/2)) * 1 = 1/(2sqrt(x + 25))
Use the quotient rule now.
f'(x) = (g'(x) * h(x) - g(x) * h'(x))/(h(x))^2
f'(x) = (3x^2(sqrt(x + 25)) - x^3/(2sqrt(x + 25)))/(sqrt(x + 25))^2
f'(x) = ((6x^2(x + 25) - x^3)/(2sqrt(x + 25)))/(x +25)
f'(x) = (5x^3 + 150x^2)/(2(x+ 25)^(3/2))
There will be critical points whenever the derivative equals 0 or is undefined. The derivative is undefined at x = -25. Set the derivative to 0 to find the other critical points.
0 = (5x^3 + 150x^2)/(2(x + 25)^(3/2))
0 = 5x^3 + 150x^2
0 = 5x^2(x + 30)
x = 0 and -30
So, there will be additional critical points at x = 0 and x= -30.
However, to be a critical number, the function has to be defined at the given point. Therefore, x = -30 and x= -25 are not critical numbers. x = 0 is the only critical number.
Hopefully this helps!
