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Question
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Options
-128
-126
-120
none of these
Solution
`-128`
Given:-
`f(x)=2x^3-21x^2+36x-20`
`rArrf'(x)=6x^2-42x+36`
For a local maxima or a local minima, we must have
`f'(x)=0`
`rArr6x^2-42x+36=0`
`rArrx^2-7x+6=0`
`rArr(x-1)(x-6)=0`
`rArrx=1, 6`
Now,
`f''(x)=12x-42`
`rArrf''(1)=12-42=-30<0`
So, x = 1 is a local maxima.
Also,
`f''(6)=72-42=30>0`
So, x = 6 is a local minima.
The local minimum value is given by
`f(6)=2(6)^3-21(6)^2+36(6)-20=-128`
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