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MHT CET202311 May 2023Evening ShiftMathematicsApplication of DerivativesActual

If the function f is given by f (x)=x^3-3( a -2) x^2+3 a x+7 , for some a R , is increasing in (0,1] and decreasing in [1,5) , then a root of the equation f (x)-14 (x-1)^2 =0(x 1) is

Options

  1. A-7
  2. B6
  3. C7
  4. D5

Correct answer

C. 7

Step-by-step solution

f(x)=x^3-3(a-2) x^2+3 a x+7 As f (x) is increasing in (0,1] and decreasing in [1,5) , we get that f (x) has critical point at x=1 f ^ (1)=0 f ^ (x)=3 x^2-6( a -2) x+3 a aligned & 3(1)^2-6(a-2)+3 a=0 & a =5 & f (x)-14 (x-1)^2 = x^3-9 x^2+15 x-7 (x-1)^2 & = (x-1)^2(x-7) (x-1)^2 & =x-7 & aligned The required root is 7 .

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