# How do you find local maximum value of f using the first and second derivative tests: #f(x)=x+sqrt(2-x)#?

See below.

(Some do not apply the term "critical number" at an endpoint of the domain.)

Using the First Derivative Test

(Some do not apply the term "local minimum" at an endpoint of the domain. They would disagree with my answer about that.)

Using the Second Derivative Test (for local extrema)

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To find the local maximum value of ( f(x) = x + \sqrt{2 - x} ) using the first and second derivative tests:

- Find the first derivative of ( f(x) ) and set it equal to zero to find critical points.
- Find the second derivative of ( f(x) ) to determine the concavity of the function at the critical points.
- Use the first derivative test to determine whether the critical points are local maxima, minima, or neither.
- Use the second derivative test to confirm the nature of the critical points found in step 3.

First derivative of ( f(x) ): [ f'(x) = 1 - \frac{1}{2\sqrt{2 - x}} ]

Set ( f'(x) = 0 ) to find critical points: [ 1 - \frac{1}{2\sqrt{2 - x}} = 0 ] [ \frac{1}{2\sqrt{2 - x}} = 1 ] [ \sqrt{2 - x} = 2 ] [ 2 - x = 4 ] [ x = -2 ]

Second derivative of ( f(x) ): [ f''(x) = \frac{1}{4(2 - x)^{3/2}} ]

Substitute ( x = -2 ) into ( f''(x) ) to determine the concavity: [ f''(-2) = \frac{1}{4(2 - (-2))^{3/2}} = \frac{1}{4(4)^{3/2}} > 0 ]

Since ( f''(-2) > 0 ), the function is concave up at ( x = -2 ), indicating a local minimum.

Therefore, the local maximum value of ( f(x) ) does not occur at ( x = -2 ). There is no local maximum value 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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