\(1.\) 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. Hence, the second derivative is positive, i.e.
\(\mathrm{f}''(x) &gt; 0\)
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. Hence, the second derivative is negative, i.e.
\(\mathrm{f}''(x) &lt; 0\)
For example: \(y = \ln x\) is concave downwards for all \(x &gt; 0\).
\(2.\) Point of Inflexion
A point of inflexion is a point where the curve changes concavity, either from up to down or down to up. If a point on the curve of \(\mathrm{f}(x)\) is a point of inflexion, then
<ol class="bullet-list"><li><div class="list-item-content">\(\mathrm{f}''(x) = 0\), and</div></li><li><div class="list-item-content">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.</div></li></ol>
\(3.\) Types of Point of Inflexion
<ol class="bullet-list"><li><div class="list-item-paragraph">Stationary point of inflexion: \(\mathrm{f}'(x) = 0\), \(\mathrm{f}''(x) = 0\) and \(\mathrm{f}''(x)\) changes signs. For example: \(y = x^3\) at \(x = 0\).</div></li><li><div class="list-item-paragraph">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 \(x = 0\).</div></li></ol>

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. Hence, the second derivative is positive, i.e.

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

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. Hence, the second derivative is negative, i.e.

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

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

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.

**Stationary point of inflexion:** \(\mathrm{f}'(x) = 0\), \(\mathrm{f}''(x) = 0\) and \(\mathrm{f}''(x)\) changes signs.\n\nFor example: \(y = x^3\) at \(x = 0\).

**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.\n\nFor example: \(y = x^3 - x\) 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 \(x = 0\).

Concavity of Curves and Point of Inflexion (Optional)

Concavity of Curves and Point of Inflexion (Optional)