KCET2024MathematicsApplication of Derivatives

The value of C in (0,2) satisfying the mean value theorem for the function f(x)=x(x-1)^2, x [0,2] is equal to

Options

A3 / 4
B4 / 3
C1 / 3
D2 / 3

Correct answer

B. 4 / 3

Step-by-step solution

f(x)=x(x-1)^2 ; x [0,2]f^ (c)= f(b)-f(a) b-a ; f(2)=2, f(0)=0 f^ (x)=(x-1)^2+2 x(x-1)=x^2+1-2 x+2 x^2-2 x=3 x^2-4 x+1 So, f^ (c)=3 c^2-4 c+1 Thus, 3 c^2-4 c+1= f(2)-(0) 2-0 = 2-0 2-0 =1 c(3 c-4)=0 Hence, c= 4 3

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