Increasing and decreasing functions

• Find the set of values of $x$ for which $f$ is decreasing (or increasing).
• Show that $f$ is an increasing function.

Giveaway words: increasing, decreasing, set of values.

Walk along the graph from left to right. Uphill means increasing, and the gradient is positive: $f'(x) > 0$.

Downhill means decreasing, and the gradient is negative: $f'(x) < 0$.

Same four steps every time:

1. Differentiate.
2. Write the inequality: increasing means $f'(x) > 0$, decreasing means $f'(x) < 0$.
3. Solve it. If $f'(x)$ is a quadratic, factorise it, then sketch it to see which part you need. For a U shape (positive $x^2$ term) it's below the axis between the roots and above it outside them. For an ∩ (negative $x^2$ term) it's the other way round.
4. Answer with an inequality in $x$, like $1 < x < 3$.

For "show that $f$ is increasing", step 3 is completing the square instead, and step 4 is a sentence.