# How do use the first derivative test to determine the local extrema #(x^2-10x)^4#?

The function has local (and global) minima at

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To use the first derivative test to determine the local extrema of ( f(x) = (x^2 - 10x)^4 ), follow these steps:

- Find the first derivative of ( f(x) ), denoted as ( f'(x) ).
- Set ( f'(x) ) equal to zero and solve for ( x ). These values of ( x ) are potential critical points.
- Determine the sign of ( f'(x) ) in intervals separated by the critical points.
- If ( f'(x) ) changes sign from positive to negative at a critical point, it indicates a local maximum at that point. If it changes sign from negative to positive, it indicates a local minimum.
- If ( f'(x) ) does not change sign at a critical point, the test is inconclusive for that point.

To execute these steps specifically for the function ( f(x) = (x^2 - 10x)^4 ):

- Calculate ( f'(x) ).
- Solve ( f'(x) = 0 ) for ( x ) to find critical points.
- Examine the sign of ( f'(x) ) in intervals defined by the critical points.
- Determine the behavior of ( f(x) ) at each critical point according to the first derivative test.

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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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