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 fro

Question

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:

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Correct answer: 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 diff

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:

Options

Correct answer

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