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

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Answer 1 · 2016-03-19T21:43:23

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

You will have relative maxima and minima in the values in which the derivate of the function is 0.

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

Assuming that we are dealing with real numbers, the zeros of the derivate will be:

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

Now we must calculate the second derivate to see what kind of extreme these values correspond:

#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 occurs at

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

No other maxima or minima exist, so this one is also an absolute minimum.