# What are the extrema of #f(x)=x^3-2x+5 # on #[-2,2]?

Minimum:

Maximum:

Steps:

Here is the graph for verification purposes: graph{x^3-2x+5 [-6.084, 6.4, 1.095, 7.335]}

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To find the extrema of ( f(x) = x^3 - 2x + 5 ) on the interval ([-2, 2]), we first find the critical points by taking the derivative of the function and setting it equal to zero. Then, we check the values of the function at the critical points and the endpoints of the interval.

- Find the derivative of ( f(x) ): ( f'(x) = 3x^2 - 2 ).
- Set ( f'(x) = 0 ) to find critical points: ( 3x^2 - 2 = 0 ).
- Solve for ( x ): ( x^2 = \frac{2}{3} ).
- Critical points: ( x = \pm \sqrt{\frac{2}{3}} ).
- Evaluate ( f(x) ) at the critical points and endpoints:
- ( f(-2) = -13 )
- ( f(2) = 11 )
- ( f\left(-\sqrt{\frac{2}{3}}\right) \approx 5.63 )
- ( f\left(\sqrt{\frac{2}{3}}\right) \approx 5.63 )

- Identify the minimum and maximum values within the given interval:
- Minimum value: ( -13 ) at ( x = -2 )
- Maximum value: ( 11 ) at ( x = 2 )

Therefore, the minimum value of ( f(x) ) on ([-2, 2]) is ( -13 ) at ( x = -2 ), and the maximum value is ( 11 ) at ( x = 2 ).

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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.

- (a) Find the interval on which f is increasing and decreasing? (b) Find the local maximum value of f? (c) Find the inflection point? (d) Find the interval on which f is concave up and concave down?
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- What are the local extrema, if any, of #f (x) =(x^2-2x)^3+(4x^2-3x^4)*e^(2x)#?
- How do you find the number c that satisfies the conclusion of the Mean Value Theorem for #f( x) = e^(-2x)# on [0,3]?

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