Question

Consider the function f(x) = |x - 2| + |x - 5|, x ∈ R.

Statement 1: f′(4) = 0.

Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).

(A) Statement−1 is false, statement−2 is true.

(B) Statement−1 is true; statement−2 is true; statement 2 is a correct explanation for statement−1.

(C) Statement−1 is true; statement−2 is true; statement 2 is not a correct explanation for statement−1.

(D) Statement−1 is true, statement−2 is false.

Answer 1

Answer is (B) Statement−1 is true; statement−2 is true; statement 2 is a correct explanation for statement−1.

f(x) = 7 - 2x; x > 2

= 3; 2 ≤ x ≤ 5

= 2x - 7; x > 5

f(x) is constant function in [2, 5]. f is also continuous in [2, 5] and differentiable in (2, 5) and f(2) = f(5); by Rolle’s theorem f ′(4) = 0. Therefore, both Statement 2 and Statement 1 are true and Statement 2 is correct explanation for Statement−1.