For the function f(x) = e^ |x| - |x|, x R , consider the following statements: Statement I: f is differentiable for all x R . Statement II: f is increasing in (- , - 2 ). In the light of the above statements, choose the correct answer from the options given below:
A. Both Statement I and Statement II are true
For Statement I: The given function is f(x) = e^ |x| - |x|. For x > 0, f(x) = e^ x - x. Differentiating with respect to x, we get: f'(x) = e^ x x - 1 The right-hand derivative at x = 0 is: f'(0^+) = e^ 0 0 - 1 = 1(1) - 1 = 0 For x Differentiating with respect to x, we get: f'(x) = -e^ - x x + 1 The left-hand derivative at x = 0 is: f'(0^-) = -e^ - 0 0 + 1 = -1(1) + 1 = 0 Since f'(0^+) = f'(0^-) = 0, the function f(x) is differentiable at x = 0. For all other x R (x 0), the function is a composition of differentiable functions and is therefore differentiable. Thus, Statement I is true. For Statement II: We need to check the monotonicity of f(x) in the interval (- , - 2 ). Since x f'(x) = 1 - e^ - x x In the interval (- , - 2 ) (which lies in the third quadrant), x Since e^ - x > 0 and x 0. Therefore, f'(x) = 1 - e^ - x x > 1 > 0. Since f'(x) > 0, the function f(x) is strictly increasing in (- , - 2 ). Thus, Statement II is true. Both Statement I and Statement II are true. Answer: Both Statement I and Statement II are true
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