Concavity of Curves and Point of Inflexion (Optional)

\(1.\) Concavity of Curves

Concavity tells us how a curve bends.

Concave Upwards

The curve of \(\mathrm{f}(x)\) bends upwards like a cup (\(\cup\)), either decreasing at a decreasing rate or increasing at an increasing rate. The gradient of the curve of \(\mathrm{f}(x)\) is increasing, i.e., \(\mathrm{f}'(x)\) is increasing. A sufficient test for strict upward concavity is a positive second derivative:

\(\mathrm{f}''(x) > 0\)

For example: \(y = \mathrm{e}^x\) is concave upwards for all \(x \in \mathbb{R}\).

Concave Downwards

The curve bends downwards like a cap (\(\cap\)), either increasing at a decreasing rate or decreasing at an increasing rate. The gradient of the curve of \(\mathrm{f}(x)\) is decreasing, i.e., \(\mathrm{f}'(x)\) is decreasing. A sufficient test for strict downward concavity is a negative second derivative:

\(\mathrm{f}''(x) < 0\)

For example: \(y = \ln x\) is concave downwards for all \(x > 0\).

\(2.\) Point of Inflexion

A point of inflexion is a point where the curve changes concavity, either from up to down or down to up. For a curve twice differentiable near the point, if a point on the curve of \(\mathrm{f}(x)\) is a point of inflexion, then

• \(\mathrm{f}''(x) = 0\), and
• The values of \(\mathrm{f}''(x)\) on both sides of the point are different, i.e. \(\mathrm{f}''(x)\) changes signs at the point of inflexion.

\(3.\) Types of Point of Inflexion

• Stationary point of inflexion: \(\mathrm{f}'(x) = 0\), \(\mathrm{f}''(x) = 0\) and \(\mathrm{f}''(x)\) changes signs.
  For example: \(y = x^3\) at \(x = 0\).
• Non-stationary point of inflexion: \(\mathrm{f}'(x) \neq 0\), \(\mathrm{f}''(x) = 0\) and \(\mathrm{f}''(x)\) changes signs.
  For example: \(y = x^3 - x\) at \(x = 0\).

The definition of inflexion is a change in concavity. The displayed second-derivative criteria concern the differentiable examples here; at a general inflexion point the second derivative need not exist. Even for a strictly convex curve it can vanish at an isolated point, for example \(y=x^4\) at 0. A zero second derivative alone proves no change in concavity.