MCQ(Mean Theorem)

1) If f(x)= Ax² + Bx + C is such that f(a)= f(b), then find the value of c in Rolle's theorem

A) a B) b C) a+ b D) (a+ b)/2.

2) n is a positive integer. If the value of c prescribed in Rolle's theorem for the function f(x)= 2x( x- 3)ⁿ on the interval [0, 2√3] is 3/4, find the value of n

A) 1 B) 2 C) 0 D) 3.

3) Find the value of c prescribed by Lagrange's mean value theorem for the function f(x)= √(x² - 4) defined in (2,3).

A) √2 B) √3 C) √4 D) √5

4) For the function f(x)= x + 1/x, x belongs to (1,3), the value of c for the Lagrange's mean value theorem is

A) 1 B) √3. C) 2 D) none

5) If from Lagrange's mean value theorem, we have

f'(x)= (f(b) - f(a))/(b - a), then

A) a< x ≤ b B) a≤ x < b

C) a< x < b. D) a≤ x ≤ b

6) The value of c in Rolle's theorem when f(x)= 2x³ - 5x² - 4x + 3, x belongs to (1/3, 3) is

A) 2. B) -1/3 C) -2 D) 2/3

7) When the tangent to the curve y= x log x is parallel to the chord joining the points (1,0) and (e,e), the value of x is

A) ₑ 1/(1- e). B) ₑ(e - 1)

C) ₑ (2e - 1/(e - 1) D) (e -1)/e

8) The value of c in Rolle's theorem for the function f(x)={x(x +1)/eˣ} defined on (-1,0) is

A) 0.5 B) (1+ √5)/2 C) (1- √5)/2. D) - 0.5

9) The value of c in Lagrange's mean value theorem for the function f(x)= x(x - 2) when x belongs to (1,2) is

A) 1 B) 1/2 C) 2/3 D) 3/2.

10) The value of c in Rolle's theorem for the function f(x)=x³ - 3x in the interval (0, √3) is

A) 1. B) -1 C) 3/2 D) 1/3

11) If f(x)= eˣ sinx in (0,π), then c in Rolle's theorem is

A) π/6 B) π/4 C) π/2 D) 3π/4.