# What are the local extrema of #f(x)= x^3-7x#?

The corresponding y values may be found by substituting back into the original equation.

The graph of the function makes verifies the above calculations.

graph{x^3-7x [-16.01, 16.02, -8.01, 8]}

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To find the local extrema of ( f(x) = x^3 - 7x ), we first find its derivative, ( f'(x) ), and then solve for points where ( f'(x) = 0 ) or where ( f'(x) ) is undefined.

[ f'(x) = 3x^2 - 7 ]

Setting ( f'(x) ) equal to zero and solving for ( x ), we get:

[ 3x^2 - 7 = 0 ] [ 3x^2 = 7 ] [ x^2 = \frac{7}{3} ] [ x = \pm \sqrt{\frac{7}{3}} ]

The critical points are ( x = -\sqrt{\frac{7}{3}} ) and ( x = \sqrt{\frac{7}{3}} ).

To determine the nature of the extrema, we examine the sign of the derivative around these critical points.

When ( x < -\sqrt{\frac{7}{3}} ), ( f'(x) ) is positive, indicating ( f(x) ) is increasing. When ( -\sqrt{\frac{7}{3}} < x < \sqrt{\frac{7}{3}} ), ( f'(x) ) is negative, indicating ( f(x) ) is decreasing. When ( x > \sqrt{\frac{7}{3}} ), ( f'(x) ) is positive again, indicating ( f(x) ) is increasing.

So, we have a local maximum at ( x = -\sqrt{\frac{7}{3}} ) and a local minimum at ( x = \sqrt{\frac{7}{3}} ).

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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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