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Manipal MET2015MathematicsApplication of Derivatives

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

Options

  1. Aincreasing on [- 1 2 , 1 ]
  2. Bdecreasing on R
  3. Cincreasing on R
  4. Ddecreasing on [- 1 2 , 1 ]

Correct answer

A. increasing on [- 1 2 , 1 ]

Step-by-step solution

We have, f(x)=x e^ x(1-x) On differentiating both sides w.r.t. x , we get aligned & f^ (x)=e^ x(1-x) +x(1-2 x) e^ x(1-x) & f^ (x)= (1+x-2 x^2 ) e^ x(1-x) & f^ (x)=-(x-1)(2 x+1) e^ x(1-x) aligned Since, e^ x(1-x) 0 for all x . Therefore, signs of f^ (x) for different values of x are as shown below: Clearly, f(x) is increasing on [- 1 2 , 1 ] and decreasing on (- ,- 1 2 ] [1, ) .

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