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MHT CET202311 May 2023Morning ShiftMathematicsApplication of DerivativesActual

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

Options

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

Correct answer

A. increasing in [- 1 2 , 1 ]

Step-by-step solution

aligned f (x) & =x e ^ x(1-x) f ^ (x) & =x e ^ x(1-x) [x(-1)+(1-x)]+ e ^ x(1-x) & = e ^ x(1-x) (x-2 x^2+1 ) aligned For f (x) to be increasing, f ^ (x) 0 aligned & e ^ x(1-x) (x-2 x^2+1 ) 0 & x-2 x^2+1 0 & 2 x^2-x-1 0 & (2 x+1)(x-1) 0 & x [- 1 2 , 1 ] aligned For f (x) to be decreasing, f ^ (x) 0 aligned & (2 x+1)(x-1) 0 & x (- ,- 1 2 ] [1, ) aligned

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