# How do you find the local extrema for #f(x) = 3x^5 - 10x^3 - 1# on the interval [-1,1]?

That function has no local extrema on that interval. (It does, of course, have global extrema on the closed interval.)

I take the definition of local maximum to be:

(For local minimum, reverse the inequality.)

Alternatives

If you are being graded based on your answer, check you grader's definitions of local extrema.

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To find the local extrema of (f(x) = 3x^5 - 10x^3 - 1) on the interval ([-1,1]), follow these steps:

- Find the critical points by setting the derivative equal to zero and solving for (x).
- Evaluate (f(x)) at these critical points and at the endpoints of the interval.
- The maximum and minimum values within the interval will be the local extrema.

Now, let's proceed with these steps:

- Find the derivative of (f(x)):

[ f'(x) = 15x^4 - 30x^2 ]

Set (f'(x) = 0) and solve for (x):

[ 15x^4 - 30x^2 = 0 ]

[ 15x^2(x^2 - 2) = 0 ]

[ x = 0, \pm \sqrt{2} ]

- Evaluate (f(x)) at the critical points and endpoints:

For (x = 0), (f(0) = -1).

For (x = \pm \sqrt{2}), calculate (f(\sqrt{2})) and (f(-\sqrt{2})).

For (x = 1), (f(1) = -8).

For (x = -1), (f(-1) = -12).

- Compare these values to determine the local extrema.

Local maximum: (f(-1) = -12)

Local minimum: (f(\sqrt{2}))

Local maximum: (f(0) = -1)

Local minimum: (f(1) = -8)

So, the local extrema for (f(x)) on the interval ([-1,1]) are:

Local maximum at (x = -1) with a value of -12.

Local minimum at (x = \sqrt{2}) with a value to be calculated.

Local maximum at (x = 0) with a value of -1.

Local minimum at (x = 1) with a value of -8.

Now, calculate (f(\sqrt{2})):

[ f(\sqrt{2}) = 3(\sqrt{2})^5 - 10(\sqrt{2})^3 - 1 ]

[ f(\sqrt{2}) = 24\sqrt{2} - 20\sqrt{2} - 1 ]

[ f(\sqrt{2}) = 4\sqrt{2} - 1 ]

Therefore, the local minimum at (x = \sqrt{2}) has a value of (4\sqrt{2} - 1).

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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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