Question

Find the critical numbers of the function f(x)=x¹/4−x³/4

a) x= 1/4
​b) x=1
c) x=−1
d) x=− 1/4
​

Answer 1

The critical number of the function f(x) = x1/4 − x3/4 is found by differentiating and setting the derivative equal to zero. Upon inspection, we can see that x = 1 is the critical number as it is the only value that makes the derivative equal to zero.

Step-by-step explanation:

To find the critical numbers of the function f(x) = x1/4 − x3/4, we need to first determine where the derivative of the function is equal to zero or does not exist. The derivative of the function, f'(x), can be found using the power rule.

Let's differentiate the function:

f'(x) = ¼x−3/4 − ¾ x−1/4

To find the critical numbers, we set the derivative equal to zero:

0 = ¼x−3/4 − ¾ x−1/4

This equation does not have a simple analytical solution and requires further algebraic manipulation to solve for x. However, we can check the options given in the question by substituting them into the derivative to see if the result is zero.

By inspection, we can see that x = 1 is a solution since:

f'(1) = ¼(1)−3/4 − ¾(1)−1/4 = 0

Therefore, the critical number for the function is x = 1.