Increasing and decreasing functions
- A function is increasing when the curve goes up, and decreasing when it goes down, as you read it left to right. The sign of the gradient tells you which.
- Use the derivative to turn the question into an inequality. You already know how to solve those.
Increasing means (the graph goes up). Decreasing means (it goes down). So “find where it is increasing” just means “solve ”.
The sign of the gradient
- A function is increasing where its graph goes up as increases (the gradient is positive). It is decreasing where it goes down (the gradient is negative).
- Differentiate, then solve the inequality. The only new skill is seeing what the question is really asking.
Worked examples
Worked example
Find the values of for which is decreasing
- Differentiate: .
- Decreasing means this is negative: .
- Rearrange: .
Answer
Worked example
Find where is increasing
From textbook WE 8.2. This is a cubic, so its derivative is a quadratic. The answer is usually two intervals.
- Differentiate and ask for : .
- Divide by and factorise: .
- This upward parabola is positive outside its roots and .
Answer
Where the marks go
Common mistake
For a cubic, the “increasing” answer is usually two intervals, not one. The derivative is a quadratic. So sketch its parabola and read off where it is positive. For an upward parabola that is outside the roots; for a downward one it is between them.
Common mistake
Take a function like . “Show it is decreasing” means show for all in the domain. A squared term on the bottom is always positive, so the sign of the derivative never changes. Explain the sign. Do not just work out one value.
Now you try
A company makes items per day. The profit is . Find the values of for which the profit is decreasing. (9709-style (textbook Ex 8A Q9))
Your turn— tap to reveal the worked answer (decreasing ⇒ P′(x) < 0)
. Divide by : .
Factorise: , negative between the roots.
Answer