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) - Mathematics MCQ AIEEE 2012 2026 | ExamTest.live

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Consider the function f (x) = ∣ x - 2∣ + ∣ x - 5∣, x \in 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 $Statement 1 is true, statement 2 is false$

Solution & Step-by-step Explanation

For x \in [2, 5] : f (x) = (x - 2) - (x - 5) = x - 2 - x + 5 = 3 .Since f (x) = 3 (a constant) for all x \in [2, 5] : f is continuous on [2, 5] . f is differentiable on (2, 5) with f^{'} (x) = 0 . f (2) = 3 and f (5) = 3 .So, Statement 2 is true.Since x = 4 is in the interval (2, 5), f^{'} (4) = 0 .Statement 1 is true. Statement 2 is the condition for Rolle's Theorem which implies the existence of c such that f^{'} (c) = 0, hence it explains Statement 1.

Try it yourself before checking the explanation above.

Consider the function f (x) = ∣ x - 2∣ + ∣ x - 5∣, x \in R .Statement 1: f^{'} (4) = 0 Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f (2) = f (5) .

Statement 1 is false, statement 2 is true

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

Statement 1 is true, statement 2 is false

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