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NDA2025MathematicsContinuity and DifferentiabilityActual

For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5] . Consider the following statements: I. The function is differentiable at x = 3 . II. The function is differentiable at x = 4 . Which of the statements given above is/are correct?

Options

  1. AI only
  2. BII only
  3. CBoth I and II
  4. DNeither I nor II

Correct answer

D. Neither I nor II

Step-by-step solution

The given function is f(x) = |x - 3| + |x - 4| defined on [0, 5] . We can redefine the function by removing the absolute value signs in different intervals: f(x) = cases -(x - 3) - (x - 4), & 0 x Simplifying this, we get: f(x) = cases 7 - 2x, & 0 x Differentiating f(x) with respect to x in the open intervals: f'(x) = cases -2, & 0 At x = 3 : Left Hand Derivative (LHD) = -2 Right Hand Derivative (RHD) = 0 Since LHD RHD, f(x) is not differentiable at x = 3 . At x = 4 : Left Hand Derivative (LHD) = 0 Right Hand Deriva

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