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

1 Answer
Mar 19, 2016

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

Explanation:

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.