How do you find critical points for #f(x) = x^4 - 8x^3 - 16x +5#?

Answer 1

Use whatever tools you have for finding irrational solution to the cubic equation: #x^3-6x^2-4 = 0#

For #f(x) = x^4 - 8x^3 - 16x +5#,
we have #f'(x) = 4x^3 - 24x^2 - 16 = 4(x^3-6x^2-4)#.

So the critical numbers are the solutions to

#x^3-6x^2-4 = 0#
Descartes' Rule of Signs assures us that there is a positive root. But none of #1, 2, 4# are roots, so the positive root is irrational.

We can also use the rule of signs to see that there are no negative roots. (So the remaining two roots are imaginary.)

Perhaps you are expected to know the general solution to the cubic, or you are using graphing technology, or you know Newton's or some other method for successive approximations to the root.

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Answer 2

To find the critical points of ( f(x) = x^4 - 8x^3 - 16x + 5 ), follow these steps:

  1. Compute the first derivative, ( f'(x) ).
  2. Find the values of ( x ) where ( f'(x) = 0 ) or where ( f'(x) ) does not exist.
  3. These values of ( x ) are the critical points of the function.

First derivative: [ f'(x) = 4x^3 - 24x^2 - 16 ]

Now, set ( f'(x) ) equal to zero and solve for ( x ): [ 4x^3 - 24x^2 - 16 = 0 ]

This equation can be solved by various methods such as factoring, synthetic division, or numerical methods. Once you find the values of ( x ) satisfying this equation, those are the critical points of the function.

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Answer from HIX Tutor

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.

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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