Increasing & Decreasing Functions

Whether a curve is increasing (rising) or decreasing (falling) is a question about the sign of the gradient, so it reduces to solving an inequality in $\frac{dy}{dx}$.

The rule

• $\dfrac{dy}{dx} > 0$: the function is increasing (rising as $x$ increases).
• $\dfrac{dy}{dx} < 0$: the function is decreasing (falling).
• $\dfrac{dy}{dx} = 0$: a stationary point, the boundary between the two.

So to find where a curve increases, differentiate and solve $\frac{dy}{dx} > 0$.

Why it ends in a quadratic inequality

For a cubic, $\frac{dy}{dx}$ is a quadratic, so “where is it increasing?” becomes a quadratic inequality, with the stationary points as the boundaries.

For $y = x^3 - 3x^2 - 9x + 5$: $\frac{dy}{dx} = 3x^2 - 6x - 9 = 3(x-3)(x+1)$. Increasing where $\frac{dy}{dx} > 0$: outside the roots, $x < -1$ or $x > 3$. Decreasing where $\frac{dy}{dx} < 0$: between them, $-1 < x < 3$.

Find the stationary points first (where $\frac{dy}{dx} = 0$), then the regions follow.

The link to stationary points

The stationary points are exactly where the function switches between increasing and decreasing, which is why a maximum has the curve increasing before it and decreasing after, and a minimum the reverse. This connection is why the sign of $\frac{dy}{dx}$ either side of a stationary point classifies it.

Common mistakes

• Reading “increasing” as $\frac{dy}{dx} < 0$ (it’s $> 0$).
• Giving the “between the roots” region for increasing, when for an upward $\frac{dy}{dx}$ parabola that region is where it decreases.
• Stopping at $\frac{dy}{dx} = 0$ (that gives boundaries, not intervals).

Full topic context: Calculus notes, and the worked increasing/decreasing example.