Concavity and convexity ~ Easy to understand maths

When you plot a function in the graph, if the curve opens towards the positive Y-axis then it is said to be concave up (or convex down) function while if it opens downwards then the curve is said to be concave down (or convex up ) function.

Concavity and convexity

Consider the curve y=f(x) in [ a,b ]. Let it be continuous and possessing tangent at every point in ( a,b ).
Draw a tangent at any point P ( c,f(c) ) on the curve. Let us assume that this tangent is not parallel to Y-axis so that f'(c) is some finite number.

Now there are three mutually exclusive possibilities to consider.

1. Concave upward

A portion of the curve on both side of P, however small it may be, lies above the tangent at P ( i.e. towards the +ve direction of Y-axis ). In this case, we say that the curve is concave upward or convex downward at P. Such curve "hold water".
As x-increase, f'(x) is either of the same sign and increasing or change sign from -ve to +ve. In either case, the slop f'(x) is increasing and f''(x) >0. Such graphs are bending upward or the portion lies above the chord.

2. Concave downward

A portion of the curve on both side of P, however small it may be, lies below the tangent at P ( i.e. towards the -ve direction of Y-axis ). In this case, we say that the curve is convex upward or concave downward at P.
As x-increase, f'(x) is either of the same sign and decreasing or change sign from +ve to -ve. In either case, the slop f'(x) is decreasing and f''(x) <0. Such graphs are bending downward or the portion lies below the chord.

3. Point of inflection

The portion of the curve on the two side of P lies on different side of the tangent at P i.e, the curve crosses the tangent at P. In this case we say that P is a point of inflection on the curve.
So, at a point of inflection the curve change from concave upward so convex downward or vice-versa.

So at a point of inflection f''(x)=0.

☆Definition of concavity and convexity

A curve is said to be concave downward ( or convex upward ) on the interval ( a,b ) if all the points of the curve lies below any tangent to it on that interval. It is said to be concave upward ( or convex downwards ) on the interval ( a,b ) if all the points of the curve lies above any tangent to it on that interval.

☆Working method for concavity and convexity

(1). Evaluate $\frac{d^2 y}{d x^2}$

(2). Find the interval ( a,b ) for which $\frac{d^2 y}{d x^2}>0$ .
Then ( a,b ) is the interval of being convex downward.

(3). Find the interval ( a,b ) for which $\frac{d^2 y}{d x^2}<0$ .
Then ( a,b ) is the interval of being convex upward.

Examples

Q1. Prove that the curve $y=e^x$ is concave upward for all x∈ℝ.
Sol.
Here $y=e^x$
$\therefore \: \: \: \: \: \: \frac{dy}{dx}=e^x$
and $\frac{d^2y}{dx^2}=e^x$

$\because \: \: \frac{d^2y}{dx^2}>0 \: for \: all\; x\epsilon \mathbb{R}$
The curve $y=e^x$ is concave upward $\forall \: \: x\: \epsilon \: \mathbb{R}.$

Q2. Prove that the curve $y=\log x$ is everywhere concave downward for x >0.
Sol.
Here $y=\log x$ , x >0
$\therefore \: \: \: \; \frac{dy}{dx}=\frac{1}{x},\: x> 0$
$\frac{d^2y}{dx^2}=-\frac{1}{x^2},\: x> 0$
Now $\frac{d^2y}{dx^2}> 0\: \; \forall \: x>0$
Hence the curve is concave downward or convex upward for x >0.

Wednesday, 20 February 2019