Find the critical numbers of the function
f(x) = x^{1/4}-x^{-3/4}.
f' = (x+3) / 4x^(7/4)
f' = 0 when x = -3.
what about x=0?
To find the critical numbers of a function, we need to find the points where the derivative of the function is equal to zero or undefined.
Let's start by finding the derivative of the function `f(x) = x^(1/4) - x^(-3/4)`.
To find the derivative, we can differentiate each term separately using the power rule of differentiation.
The derivative of `x^(1/4)` can be found using the power rule as: `(1/4)x^(-3/4)`.
Similarly, the derivative of `x^(-3/4)` can be found as: `(-3/4)x^(-7/4)`.
Now, let's find the derivative of the entire function f(x):
f'(x) = (1/4)x^(-3/4) - (3/4)x^(-7/4)
Now, to find the critical numbers, we set the derivative `f'(x)` equal to zero and solve for x:
(1/4)x^(-3/4) - (3/4)x^(-7/4) = 0
Multiplying through by 4 to eliminate the fractions:
x^(-3/4) - 3x^(-7/4) = 0
Now, let's multiply both sides by `x^(7/4)` to eliminate the negative exponent:
x^(7/4) * x^(-3/4) - 3 * x^(7/4) * x^(-7/4) = 0
x^(4/4) - 3x^(0) = 0
Simplifying further:
x - 3 = 0
x = 3
Therefore, the critical number of the function f(x) = x^(1/4) - x^(-3/4) is x = 3.
To find the critical numbers of a function, we need to find the values of x where the derivative of the function is either zero or undefined.
First, let's find the derivative of the function f(x) = x^(1/4) - x^(-3/4).
To differentiate each term separately, we can use the power rule.
The power rule states that the derivative of x^n (where n is any real number) is nx^(n-1).
So, the derivative of the first term x^(1/4) is:
(1/4)x^(1/4 - 1)
= (1/4)x^(-3/4).
Similarly, the derivative of the second term x^(-3/4) is:
(-3/4)x^(-3/4 - 1)
= (-3/4)x^(-7/4).
Now, let's set the derivative equal to zero and solve for x:
(1/4)x^(-3/4) - (-3/4)x^(-7/4) = 0.
Multiplying through by 4 to eliminate the fractions, we get:
x^(-3/4) + 3x^(-7/4) = 0.
To simplify this equation, let's multiply through by x^(7/4) to get rid of the negative exponents:
x^(7/4)x^(-3/4) + 3x^(7/4)x^(-7/4) = 0.
This becomes:
x^4 + 3 = 0.
At this point, we realize that there are no real solutions to this equation since x^4 is always positive and adding 3 to it will always result in a positive number. Therefore, the derivative is never zero for any value of x, so there are no critical numbers for the function f(x) = x^(1/4) - x^(-3/4).