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Solution


Q.

The function f(x)=2x+3(x)2/3,  x∈R, has                                   [2024]
1 exactly one point of local minima and no point of local maxima  
2 exactly two points of local maxima and exactly one point of local minima  
3 exactly one point of local maxima and no point of local minima  
4 exactly one point of local maxima and exactly one point of local minima  

Ans.

(4)

We have, f(x)=2x+3x2/3

⇒f'(x)=2+3·23x-1/3=2(1+x-1/3)

For critical points, f'(x)=0

⇒1+x-1/3=0⇒x1/3=-1⇒x=-1

The sign scheme of f'(x) is

∴  x=-1 is the point of local maxima and x=0 is the point of local minima.