Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x)=x^(4/5)(x − 6)^2
by the product rule ....
f ' (x) = x^(4/5) (2)(x-6) + (4/5)x^(-1/5) (x-6)^2
= (-1/5)x^(-1/5) (x-6) [10x + 4(x-6) ]
= (-2/5)x^(-1/5) (x-6) (7x-12)
= 0 for critical numbers (max,min)
-1/5x^(-1/5) = 0 ---> no solution
x-6 = 0 ---> x = 6
7x-12=0 ---> x = 12/7
take it from there
To find the critical numbers of a function, we need to locate the x-values where the derivative of the function is either zero or undefined. The critical numbers help us identify the potential local extrema (maximum or minimum) points of the function.
First, let's find the derivative of the function f(x) with respect to x using the product rule and the power rule.
f(x) = x^(4/5)(x - 6)^2
Using the product rule, the derivative f'(x) is given by:
f'(x) = [ (4/5)x^(-1/5) *(x - 6)^2 ] + [ x^(4/5) * 2(x - 6) ]
To simplify, we distribute and combine the terms:
f'(x) = (4/5)x^(-1/5)(x - 6)^2 + 2x^(4/5)(x - 6)
Now, we need to determine when the derivative f'(x) is equal to zero or undefined.
Setting f'(x) equal to zero and solving for x:
(4/5)x^(-1/5)(x - 6)^2 + 2x^(4/5)(x - 6) = 0
To solve this equation, we can multiply both sides by 5 to eliminate the fraction:
4(x - 6)^2 * x^(-1/5) + 10x^(4/5)(x - 6) = 0
We can further simplify by dividing both sides by x^(4/5) * (x - 6):
4(x - 6) / x^(1/5) + 10(x - 6) = 0
Now, let's combine like terms:
4(x - 6) + 10(x - 6) = 0
Simplifying further:
4x - 24 + 10x - 60 = 0
14x - 84 = 0
14x = 84
x = 6
So, we have found one critical number which is x = 6.
To check if x = 6 makes f'(x) undefined, we need to examine if x = 6 results in a denominator equal to zero in the derivative function f'(x). From the derivative, we notice that there are no denominators involving x, which means f'(x) is defined for all values of x.
Therefore, the critical number of the function f(x)=x^(4/5)(x − 6)^2 is x = 6.