The sign of $\mathrm{f}'$ tells whether the function rises or falls. The sign of $\mathrm{f}''$ tells whether the gradient is increasing or decreasing. Equal horizontal increments $\Delta x$ make the vertical changes directly comparable. Decreasing at a decreasing rate The curve slopes downwards. It becomes flatter as $x$ increases. $\mathrm{f}'(x)<0$ and $\mathrm{f}''(x)>0$. Increasing at an increasing rate The curve slopes upwards. It becomes steeper as $x$ increases. $\mathrm{f}'(x)>0$ and $\mathrm{f}''(x)>0$. Increasing at a decreasing rate The curve slopes upwards. It becomes flatter as $x$ increases. $\mathrm{f}'(x)>0$ and $\mathrm{f}''(x)<0$. Decreasing at an increasing rate The curve slopes downwards. It becomes steeper as $x$ increases. $\mathrm{f}'(x)<0$ and $\mathrm{f}''(x)<0$.

Describing Rates of Change

The sign of $\mathrm{f}'$ tells whether the function rises or falls. The sign of $\mathrm{f}''$ tells whether the gradient is increasing or decreasing. Equal horizontal increments $\Delta x$ make the vertical changes directly comparable.

Decreasing at a decreasing rate

The curve slopes downwards.
It becomes flatter as $x$ increases.
$\mathrm{f}'(x)<0$ and $\mathrm{f}''(x)>0$.

Increasing at an increasing rate

The curve slopes upwards.
It becomes steeper as $x$ increases.
$\mathrm{f}'(x)>0$ and $\mathrm{f}''(x)>0$.

Increasing at a decreasing rate

The curve slopes upwards.
It becomes flatter as $x$ increases.
$\mathrm{f}'(x)>0$ and $\mathrm{f}''(x)<0$.

Decreasing at an increasing rate

The curve slopes downwards.
It becomes steeper as $x$ increases.
$\mathrm{f}'(x)<0$ and $\mathrm{f}''(x)<0$.

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