Find critical Values

Question

Find all critical values for the function f(x)=x ^5/11(x-5)^2 and then list them

Answer 1

You need to compute the first derivative and then study its sign.

x = 0 is a critical point and, specifically, is an inflection point

x = 25/3 is a relative minimum

Answer 2

f(x) = x⁵ / (11(x - 5)²) = (1/11) [ x⁵/(x-5)²]

f'(x) = (1/11) [ 5x⁴(x-5)² - 2(x-5)(x⁵) ] / (x-5)⁴

= (1/11) [ x⁴(x-5)(5(x-5) - 2x)] / (x-5)⁴ = (1/11)x⁴(3x - 25) / (x - 5)³

f'(x) = 0 when x = 0, 25/3

Note: f'(x) is undefined at x = 5, but 5 is not in the domain of f(x). So, 5 is not a critical value.

Critical numbers are 0 and 25/3.