Exams · UNT (Kazakhstan), mathematics
The greatest and least value
The derivative helps find where a function is largest and smallest.
How to solve it
The greatest and least value on a segment
You need the greatest or least value of a function on a segment, for example $y = x^3 - 3x - 7$ on $[0, 2]$. A function reaches its extreme values either at the ends of the segment or where its derivative is zero. So a few points are enough to check.
Step by step
- Find the derivative. The rule $(x^n)' = nx^{n - 1}$ is usually enough: $(x^3 - 3x - 7)' = 3x^2 - 3$.
- Solve $y' = 0$: these are the critical points, $x = \pm 1$.
- Keep only those inside the segment: here $x = 1$.
- Compute the function at these points and at the ends: $y(0) = -7$, $y(1) = -9$, $y(2) = -5$.
- The largest of these numbers is the greatest value, the smallest is the least: here $-5$ and $-9$.
Common mistakes
- Forgetting the values at the ends, although the answer is often there.
- Taking a critical point that lies outside the segment.
- Answering with the point $x$ instead of the value of the function. The question asks for $y$.