How do you find the exact relative maximum and minimum of the polynomial function of #4x^8 - 8x^3+18#?

Answer 1

Only an absolute minimum at #(root(5)(3/4), 13.7926682045768......)#

Relative maxima and minima will exist in the values where the function's derivate is zero.

#f'(x)=32x^7-24x^2=8x^2(4x^5-3)#

The zeros of the derivate, assuming we are working with real numbers, will be:

# 0 and root(5)(3/4)#

To determine what kind of extreme these values correspond to, we must now compute the second derivative:

#f'(x)=224x^6-48x=16x(14x^5-3)#
#f''(0)=0 #-> inflection point
#f''(root(5)(3/4))=16root(5)(3/4)(14xx(3/4)-3)=120root(5)(3/4)>0#-> relative minimum

which takes place at

#f(root(5)(3/4))=13.7926682045768......#

This is also an absolute minimum since there are no other maxima or minima.

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Answer 2

To find the exact relative maximum and minimum of the polynomial function (4x^8 - 8x^3+18), you first need to find its critical points. Critical points occur where the derivative of the function is equal to zero or undefined.

  1. Take the derivative of the function (f(x) = 4x^8 - 8x^3+18) to find the critical points.
  2. Set the derivative equal to zero and solve for (x).
  3. Once you have the critical points, use the second derivative test to determine whether each critical point corresponds to a relative maximum, minimum, or neither.
  4. Evaluate the function at each critical point and compare the values to determine which are relative maximums or minimums.

If you follow these steps, you can find the exact relative maximum and minimum of the given polynomial 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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