Let f:(0, ) R be defined by f(x) = | _ e x| - k|x - 1| , where k is a real constant. If f is differentiable at x = 1 , then which of the following is correct?

Step-by-step solution

For x > 1 , x - 1 > 0 and _ e x > 0 . Thus, f(x) = _ e x - k(x - 1) . Differentiating with respect to x , f'(x) = 1 x - k . The right-hand derivative at x = 1 is _ x 1^+ f'(x) = 1 - k . For 0 Thus, f(x) = - _ e x + k(x - 1) . Differentiating with respect to x , f'(x) = - 1 x + k . The left-hand derivative at x = 1 is _ x 1^- f'(x) = -1 + k . Since f(x) is differentiable at x = 1 , the left-hand derivative must equal the right-hand derivative. 1 - k = -1 + k 2k = 2 k = 1 . Substituting k = 1 back into the derivative