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Manipal MET2014MathematicsApplication of Derivatives

What are the values of c for which Rolle's theorem for the function f(x)=x^3-3 x^2+2 x in the interval [0,2] is verified?

Options

  1. Ac= 1
  2. Bc=1 1 3
  3. Cc= 2
  4. DNone of these

Correct answer

B. c=1 1 3

Step-by-step solution

Here, we observe that (a) f(x) is a polynomial, so it is continuous in the interval [0,2] . (b) f^ (x)=3 x^2-6 x+2 exists for all x (0,2) . So, f(x) is differentiable for all x (0,2) and (c) f(0)=0, f(2)=2^3-3(2)^2+2(2)=0 f(0)=f(2) Thus, all the three conditions of Rolle's theorem are satisfied. So, there must exist c [0,2] such that f^ (c)=0 array ll & f^ (c)=3 c^2-6 c+2=0 & c=1 1 3 [0,2] array

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