# How do you find the critical points for #f(x) = 2x^(2/3) - 5x^(4/3)#?

You calculate its first derivative and check to see where it is equal to zero or undefined.

By definition, a critical point of a function that can be differentiated on its domain is any point where the first derivative of said function is either zero or undefined.

So, differentiate the function to get

Now, notice that one of the terms has a negative exponent. This means that you can write it is

This is equivalent to

You can further simplify thisequation to get

graph{2x^(2/3) - 5x^(4/3) [-4.933, 4.93, -2.466, 2.466]}

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To find the critical points of ( f(x) = 2x^{\frac{2}{3}} - 5x^{\frac{4}{3}} ), we first find its derivative, then set it equal to zero and solve for ( x ).

[ f'(x) = \frac{d}{dx}(2x^{\frac{2}{3}}) - \frac{d}{dx}(5x^{\frac{4}{3}}) ]

Using the power rule for differentiation, we get:

[ f'(x) = \frac{4}{3}x^{-\frac{1}{3}} - \frac{20}{3}x^{\frac{1}{3}} ]

Setting ( f'(x) ) equal to zero:

[ \frac{4}{3}x^{-\frac{1}{3}} - \frac{20}{3}x^{\frac{1}{3}} = 0 ]

Solving for ( x ), we find:

[ x^{-\frac{1}{3}} = \frac{20}{4}x^{\frac{1}{3}} ]

[ x^{-\frac{1}{3}} = 5x^{\frac{1}{3}} ]

[ x^{-\frac{1}{3}} - 5x^{\frac{1}{3}} = 0 ]

[ x^{\frac{1}{3}}(x^{-\frac{2}{3}} - 5) = 0 ]

This equation has solutions when either ( x^{\frac{1}{3}} = 0 ) or ( x^{-\frac{2}{3}} - 5 = 0 ).

For ( x^{\frac{1}{3}} = 0 ), the solution is ( x = 0 ).

For ( x^{-\frac{2}{3}} - 5 = 0 ), solving for ( x ), we get ( x = \left(\frac{1}{125}\right)^{\frac{3}{2}} ).

So, the critical points of ( f(x) ) are ( x = 0 ) and ( x = \left(\frac{1}{125}\right)^{\frac{3}{2}} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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