## Differentiation – Turning Points of a Curve

January 26, 2012 in Additional Mathematics, Arithmetic Progression, Turning Points of A Curve by Shivakkumar Vadiveyl

## Section J Differentiation – Turning Points of a Curve

One of the applications of Differentiation is to find the maximum or minimum points of a curve, which are called the turning points.

Turning Points of a Curve

Let us take a look at a function $y=4x^{2}+2x+5$. Find the coordinates of the turning point.

$begin{array}{ccccc}& y & = & 4x^{2}+2x+5 & rightarrow A& dfrac{dy}{dx} & = & 8x+2dfrac{dy}{dx}=0Rightarrow & 0 & = & 8x+2& 8x & = & -2& x & = & -dfrac{2}{8}vspace{2mm}& x & = & -dfrac{1}{4}end{array}$

Substituting for $x=-dfrac{1}{4}$ into A.

$begin{array}{ccccc}& y & = & 4x^{2}+2x+5vspace{2mm}& & = & 4(-dfrac{1}{4})^{2}+2(-dfrac{1}{4})+5vspace{2mm}& & = & 4(dfrac{1}{16})-dfrac{2}{4}+5(dfrac{4}{4})vspace{2mm}& & = & dfrac{1}{4}-dfrac{2}{4}+dfrac{20}{4}vspace{2mm}& & = & dfrac{19}{4}vspace{2mm}& y & = & 4dfrac{3}{4}end{array}$

Therefore, the coordinate of the turning point is $(-dfrac{1}{4}$,$4dfrac{3}{4})$

In this example, we have found the turning point, but, we do not know whether the turning point is a maximum or a minimum. Refer to the pages for Exercise J which has another example of maximum / minimum points or the turning points. Exercise J also goes on to determine whether the turning point is a maximum or a minimum.