Let f : R → R be defined by f x = x 2 - 3 x - 6 x 2 + 2 x + 4 Then which of the following statements…

Let f:R→R be defined by fx=x2-3x-6x2+2x+4

Then which of the following statements is(are) TRUE?

  1. f is decreasing in the interval (-2,-1)
  2. f is increasing in the interval (1,2)
  3. f is onto
  4. Range of f is -32,2

Solution

fx=x2-3x-6x2+2x+4

f'x=x2+2x+42x-3-x2-3x-62x+2x2+2x+42

=5x2+20xx2+2x+42=5x(x+4)x2+2x+42

Hence, it is increasing in (-∞,-4]∪[0,∞)

 

Now as x→-∞ then fx=1, also, x→∞ then fx=1

f-4=16+12-616-8+4=2212=116

f0=-32

Hence, range is -32, 116, and the function is into.

Asked in: JEE Advanced 2021 (Paper 1)

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