Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.12 Convergence of Taylor series.4.14 Classification of stationary points

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4.13 Sign test for increasing or decreasing functions

Let $f(x)\,$ be a function that is suitably differentiable near $x=a\,$ . Recall that $f^{\prime}(a)\,$ equals the gradient of the tangent to the graph of $f\,$ at $(a,f(a))\,$ . The tangent is either:

(i) sloping upwards,

(ii) sloping downwards, or

(iii) horizontal.

Signs of derivative

(i) If $f^{\prime}(a)>0\,$ , then $f\,$ is increasing at $a\,$ ;

(ii) If $f^{\prime}(a)<0\,$ , then $f\,$ is decreasing at $a\,$ ; and

(iii) If $f^{\prime}(a)=0\,$ , then $f\,$ is said to have a stationary point at $a\,$ .

The derivative is often used to locate points where $f(x)\,$ is increasing or decreasing. At a stationary point $a$ of a function $f$ , the tangent to the graph of $f$ is horizontal.