List the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use.
g(x)=2x^1/3−2x^−2/3
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g'(x) = 2/3 x^(-2/3) + 4/3 x^(-5/3)
= 2/3 x^(-5/3) (x+2)
critical numbers are where g'=0 or g' is undefined.
To find the critical numbers of the function g(x) = 2x^(1/3) - 2x^(-2/3), we need to find the values of x where the derivative is equal to zero or undefined.
Let's first find the derivative of g(x).
g'(x) = d/dx (2x^(1/3) - 2x^(-2/3))
To differentiate each term, we can use the power rule:
The derivative of x^n is n * x^(n-1).
Applying the power rule, we get:
g'(x) = 1/3 * 2x^(-2/3) + 2/3 * 2x^(-5/3)
Simplifying further, we have:
g'(x) = 2/3 * (x^(-2/3) + 2x^(-5/3))
Now, let's set g'(x) equal to zero and solve for x:
2/3 * (x^(-2/3) + 2x^(-5/3)) = 0
We can eliminate the fraction by multiplying both sides of the equation by 3/2:
(x^(-2/3) + 2x^(-5/3)) = 0
Next, let's solve for x by isolating each term:
x^(-2/3) = -2x^(-5/3)
Taking the reciprocal of both sides:
x^(2/3) = -1/(2x^(5/3))
Now, raise both sides of the equation to the power of 3/2:
(x^(2/3))^(3/2) = (-1/(2x^(5/3)))^(3/2)
Simplifying:
x^1 = (-1)^3 / (2^3 * x^(5/2))
x = -1 / (2^3 * x^(5/2))
Cross-multiplying:
x * 2^3 * x^(5/2) = -1
8x^(7/2) = -1
Now, let's solve for x:
x^(7/2) = -1/8
We can take the square root of both sides:
x^(7/2)^(1/7) = (-1/8)^(1/7)
x^1 = (-1/8)^(1/7)
x = -(1/8)^(1/7)
Therefore, the critical numbers of the function g(x) = 2x^(1/3) - 2x^(-2/3) are x = -(1/8)^(1/7).
Final Answer: -(1/8)^(1/7)
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 g(x):
g'(x) = d/dx [2x^(1/3) - 2x^(-2/3)]
Using the power rule, we can differentiate each term separately:
g'(x) = (2/3) * (1/3) * x^(-2/3 - 1) - (-2/3) * (-2/3) * x^(-2/3 - 1)
Simplifying:
g'(x) = (2/9) * x^(-5/3) + (4/9) * x^(-5/3)
Now, let's find the critical numbers by setting the derivative equal to zero and solving for x:
(2/9) * x^(-5/3) + (4/9) * x^(-5/3) = 0
Combining like terms:
(6/9) * x^(-5/3) = 0
Simplifying:
(2/3) * x^(-5/3) = 0
To solve for x, we can multiply both sides by 3/2:
x^(-5/3) = 0
Since any nonzero number raised to the power of zero is 1, we can conclude that there are no values of x that satisfy this equation. Therefore, there are NO critical numbers for the function g(x) = 2x^(1/3) - 2x^(-2/3).