Understanding the Sign of the Derivative: Key to Analyze Function Behavior – Explained with Examples

If f'(x) is always negative (odd exponent) or always positive (even exponent)…

then f is one to one and therefore it has an inverse.

If f'(x) is always negative (odd exponent) or always positive (even exponent), then we can make some conclusions about the behavior of the function f(x).

1. If f'(x) is always negative, then the function f(x) is decreasing on its entire domain. This is because a negative derivative means that the function is decreasing at every point. In other words, as x increases, the value of f(x) decreases.

2. If f'(x) is always positive, then the function f(x) is increasing on its entire domain. This is because a positive derivative means that the function is increasing at every point. In other words, as x increases, the value of f(x) increases.

These conclusions are based on the Mean Value Theorem, which states that if a function is continuous and differentiable on an interval, then there exists a point within that interval where the slope of the tangent line is equal to the slope of the secant line. If the derivative is always negative (resp. positive), then the slope of the tangent line is negative (resp. positive) at every point, and so the function is decreasing (resp. increasing) everywhere on the interval.

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