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Question

# If f(x)=4x3−x2−2x+1 and g(x)=[Minf(t):0≤t≤x;0≤x≤13−x;1<x≤2] then g(14)+g(34)+g(54) has the value equal to

A
74
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B
94
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C
134
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D
52
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Solution

## The correct option is D 52Given, f(x)=4x3−x2−2x+1Taking first derivative of the function f(x) we getf′(x)=2(6x2−x−1)Putting this equals to zero we get, x=−13, 12. Taking second derivative of f(x) we get, f′′(x)=2(12x−1) this shows that f(x) has a local maxima at x=−13 and a local minima at x=12. Which means in between 0≤x≤1 f(x) has a minima at 1/2 and f(x)=1 at x=0 and f(x)=2 at x=1, also f(x)=1/4 at x=1/2.As per question we have g(x)= Min f(t):0≤t≤x,g(14) = f(14) as min of f(x) in between 0 to 1/4 is at 1/4f(14) = 1/2g(34) = f(12) because in between 0 to 3/4, we have minima at x= 1/2 which is the local minima of function as well.f(12)=1/4g(54) = 74So g(14) + g(34) + g(54) = 52

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