# How do you find all critical point and determine the min, max and inflection given #f(x)=x^4#?

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To find the critical points of ( f(x) = x^4 ), we first find the derivative:

[ f'(x) = 4x^3 ]

Critical points occur where the derivative is equal to zero or undefined. Here, the derivative is never undefined, so we set it equal to zero:

[ 4x^3 = 0 ]

This gives us ( x = 0 ) as the only critical point.

To determine whether ( x = 0 ) corresponds to a minimum, maximum, or inflection point, we use the second derivative test or examine the behavior of the function around ( x = 0 ). Since the second derivative ( f''(x) = 12x^2 ) is positive for ( x = 0 ), ( x = 0 ) corresponds to a local minimum.

There are no inflection points because the function ( f(x) = x^4 ) is a polynomial of degree 4, and it does not change concavity.

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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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