# How do you find the second derivative test to find extrema for #f(x) = 2x^2lnx-5x^2#?

See the explanation.

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To find extrema for ( f(x) = 2x^2 \ln(x) - 5x^2 ), follow these steps:

- Find the first derivative of ( f(x) ) using the product rule: ( f'(x) = 4x \ln(x) + 2x - 5 ).
- Find critical points by setting ( f'(x) ) equal to zero and solving for ( x ).
- Once you have critical points, use the second derivative test to determine whether each critical point corresponds to a relative maximum, relative minimum, or neither.
- Find the second derivative of ( f(x) ): ( f''(x) = 4\ln(x) + 6/x ).
- Evaluate ( f''(x) ) at each critical point found in step 2.
- If ( f''(x) > 0 ) at a critical point, then it corresponds to a relative minimum.
- If ( f''(x) < 0 ) at a critical point, then it corresponds to a relative maximum.
- If ( f''(x) = 0 ) or is undefined at a critical point, the test is inconclusive.

That's the process for using the second derivative test to find extrema for the given function.

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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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