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5.3 First Derivative and Function Characteristics

Increasing and Decreasing Function

  • f(x) is strictly increasing in an interval (a,b) if f^{\prime}(x)>0 for all values of x in that interval.
    In this case, the tangent to any points in the interval has a positive gradient.
  • Similarly, f(x) is strictly decreasing in an interval if f^{\prime}(x)<0.
    The tangent has a negative gradient.
  • If f(x) is said to be strictly increasing (or decreasing) without any interval specified, then f^{\prime}(x)\geq0 (or f^{\prime}(x)\leq0)for ANY values of x (or precisely the domain of f(x)).
    Note: Notice the signs here are \geq, \leq instead of >, <.
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