JEE Main2012MathematicsApplication of DerivativesActual

If f(x)=x e^ x(1-x) , x R , then f(x) is

Options

Adecreasing on [-1 / 2,1]
Bdecreasing on R
Cincreasing on [-1 / 2,1]
Dincreasing on R

Correct answer

C. increasing on [-1 / 2,1]

Step-by-step solution

aligned & f(x)=x e^ x(1-x) , x R & f^ (x)=e^ x(1-x) [1+x-2 x^2 ] aligned aligned & =-e^ x(1-x [2 x^2-x-1 ] & =-2 e^ x(1-x [ (x+ 1 2 )(x-1] ) & f^ (x)=-2 e^ x(1-x) A & where A= (x+ 1 2 )(x-1) aligned Now, exponential function is always + ve and f^ (x) will be opposite to the sign of A which is -ve in [- 1 2 , 1 ] Hence, f^ (x) is + ve in [- 1 2 , 1 ] f(x) is increasing on [- 1 2 , 1 ]

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