Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
If is a real-valued differentiable function such that f^{\prime}(x)<0 \) for all
If \( f \) is a real-valued differentiable function such that \( f(x) f^{\prime}(x)<0 \) for all real \( x, \) then
- A. \( f(x) \) must be a increasing function
- B. \( f(x) \) must be a decreasing function
- C. \( |f(x)| \) must be a increasing function
- D. \( |f(x)| \) must be a decreasing function
Step-by-step solution
Given f(x) f'(x) < 0, f and its derivative have opposite signs. Consider g(x) = |f(x)|. For f(x) ≠ 0, g'(x) = (f(x) f'(x))/|f(x)| < 0 (since numerator negative, denominator positive). Hence |f| is strictly decreasing.
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