Find the absolute maximum value of g(x)=4x3−18x2−48x+60 on the interval [0,6], if it exists. ∘60 ∘164 ∘288 ∘86 ∘ f(x) has no absolute maximum on the given interval.

Question

Find the absolute maximum value of g(x)=4x3−18x2−48x+60 on the interval [0,6], if it exists.  ∘60  ∘164  ∘288  ∘86  ∘ f(x) has no absolute maximum on the given interval.

Melinda McCombs · Accepted Answer

Step 1Given,g(x)=4x3−18x2−48x+60We have to find absolute maximum value of g(x) on the interval [0,6]Step 2When we have the function f(x)Absolute maximum value occur when f&#039;(x) = 0  and  at these values of x and  at the end points of the given interval we will calculate f(x), then maximum of f(x) will be absolute maximum.So, NowFinding g&#039;(x)g′(x)=12x2−36x−48 (because dxndx=nxn−1 and d(constant)/dx = 0 )Step 3Now making g′(x)=12x2−36x−48=012x2−36x−48=0x2−3x−4=0x2−4x+x−4=0x(x-4)+1(x-4)=0(x-4)(x+1)=0then x-4 = 0 or x=4 andx+1=0 or x=-1 (it will be discarded because it is not lying in the interval [0,6])Step 4Now findingg(0)=4×03−18×02−48×0+60=60g(4)=4×43−18×42−48×4+60=−164g(6)=4×63−18×62−48×6+60=−12So,Among all the above absolute maximum occur at x = 0 and absolute maximum value is g(0) = 60Hence 60 is the correct option  (Answer)

Find the absolute maximum value of g(x)=4x^{3}-18x^{2}-48x+60 on the interval

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