How do you find all critical numbers of the function #f(x)=x^(4/5)(x4)^2#?
Critical points are points in the domain where the derivative is equal to zero or where the derivative is not defined.
Derivative Equal to Zero:
The derivative can be found by using the power rule and the chain rule.
Now we set this equal to zero:
We get:
Derivative Not Defined:
Conclusion
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To find all critical numbers of the function ( f(x) = x^{\frac{4}{5}}(x4)^2 ), follow these steps:

Find the derivative of the function using the product rule and simplify: [ f'(x) = \frac{4}{5}x^{\frac{1}{5}}(x4)^2 + x^{\frac{4}{5}}(2(x4)) ]

Set the derivative equal to zero and solve for ( x ) to find the critical numbers: [ \frac{4}{5}x^{\frac{1}{5}}(x4)^2 + x^{\frac{4}{5}}(2(x4)) = 0 ]

Simplify the equation and solve for ( x ). This may involve factoring and solving a polynomial equation.

After solving for ( x ), check each critical number to determine if it corresponds to a maximum, minimum, or point of inflection by using the first or second derivative test.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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