# How do you find local maximum value of f using the first and second derivative tests: #f(x) = 7e^x#?

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To find local maximum values of (f(x) = 7e^x) using the first and second derivative tests, follow these steps:

- Find the first derivative of (f(x)) to get (f'(x)).
- Find the critical points by setting (f'(x) = 0) and solving for (x).
- Find the second derivative of (f(x)) to get (f''(x)).
- Use the first derivative test: If (f'(x)) changes sign from positive to negative at a critical point (x = c), then (f(c)) is a local maximum.
- Use the second derivative test: If (f''(c) < 0) at a critical point (x = c), then (f(c)) is a local maximum.

Let's apply these steps to (f(x) = 7e^x):

- The first derivative is (f'(x) = 7e^x).
- Setting (f'(x) = 0), we get (7e^x = 0), which has no real solutions since (e^x) is always positive.
- The second derivative is (f''(x) = 7e^x).
- Since there are no critical points, there are no local maximum values according to the first derivative test.
- Since there are no critical points, the second derivative test is not applicable.

Therefore, the function (f(x) = 7e^x) has no local maximum values.

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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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