Find all critical numbers for the function
𝑓(𝑥)=𝑥15−𝑥−45
and then list them (separated by commas) in the box below. If there are no critical numbers, enter None .
List of critical numbers:
To find the critical numbers of a function, we need to first find its derivative and set it equal to zero.
Given the function f(x) = x^15 - x - 45, let's find its derivative.
Taking the derivative of each term, using the power rule, we get:
f'(x) = 15x^14 - 1
Now we set the derivative equal to zero and solve for x:
15x^14 - 1 = 0
To solve this equation for x, we add 1 to both sides:
15x^14 = 1
Next, divide both sides by 15:
x^14 = 1/15
To get rid of the exponent 14, we can take the 14th root of both sides:
x = (1/15)^(1/14)
Now we have found the critical number for the function f(x) = x^15 - x - 45. Taking the 14th root of (1/15) will give us the value of x.
So the critical number for the given function is x = (1/15)^(1/14).