Maxima and minima Class 12: word problems step by step
This post explains how Class 12 maxima and minima word problems work and where students lose marks.
You will also get a board-style method, a worked example, a set of checks and a simple weekly routine for home.
Why these word problems feel hard
Maxima and minima means finding the largest or smallest value of a quantity. In Class 12 this sits in the chapter on application of derivatives. A derivative tells us how fast a quantity changes. At a peak or a valley, that rate of change is zero.
The calculus part is usually short. Many students struggle with the English part. The question describes a box, a cylinder, a wire or a fence. The student must turn those words into an equation. If the equation is wrong, every later step is wrong, even when the differentiation is perfect.
Parents in Dwarka and Gurugram often describe the same pattern: the child remembers the formulas but gets stuck at the very start. That is a setting-up problem. It can be fixed with a fixed routine, not with more formulas.
The board paper is marked step by step. A neat, ordered answer shows the examiner your thinking. Even if one calculation slips later, the earlier steps can still earn credit.
The six-step method in board format
Ask your child to write these steps in the same order every time. Repetition builds the habit, and the habit removes the panic.
- Draw and name. Sketch the figure. Name every length, such as x for the side and h for the height.
- State the quantity to optimise. Write one line saying what the letter stands for, for example that V stands for volume or S stands for surface area.
- Use the given condition. The question gives one fixed fact, such as a fixed perimeter or a fixed volume. Use it to remove one variable, so the quantity depends on one variable only.
- Differentiate. Find the first derivative and put it equal to zero. The solutions are called critical points.
- Test. Use the second derivative test. If the second derivative is positive at the point, it is a minimum. If it is negative, it is a maximum.
- Answer in words. Write the final value with its unit and say what it means for the question.
Steps 1 to 3 are the setup. Spend the most time here. Steps 4 and 5 are routine once the function is correct.
A worked example: the largest area for a fixed fence
Question: A farmer has 40 m of fencing. She wants to enclose a rectangular plot of the largest possible area. Find its dimensions.
Step 1. Let the length be x m and the breadth be y m.
Step 2. Let the area be A = xy.
Step 3. The perimeter is fixed: 2(x + y) = 40, so y = 20 − x. Then A = x(20 − x) = 20x − x2.
Step 4. dA/dx = 20 − 2x. Put it equal to zero: x = 10.
Step 5. d2A/dx2 = −2, which is negative. So x = 10 gives a maximum.
Step 6. Then y = 10. The plot should be a square of side 10 m, and the largest area is 100 m2.
Notice what the student must still do after finding x. Find y, state the area and name the shape. Many answers stop at "x = 10", which is incomplete.
Common mistakes and how to correct them
| Mistake | Why it costs marks | Fix |
|---|---|---|
| Two variables left in the function | This chapter differentiates a function of one variable only | Use the given condition to remove one variable before Step 4 |
| No second derivative test | The point may be a minimum, not a maximum | Write the sign of the second derivative and say "maximum" or "minimum" |
| Wrong or missing unit | The final answer is incomplete | Write cm, m2 or cm3 in the last line |
| Ignoring the allowed range of the variable | A length cannot be negative or zero | Reject any critical point that makes a length negative |
| Answering only for x | The question asks for the dimensions or the value | Re-read the question and answer exactly what it asks |
Keep this table beside the notebook. When a problem goes wrong, find the row that matches and note it down.
How to verify an answer before moving on
A student should never hand over a solution without a quick check. Three checks take under a minute.
- Substitute back. Put the value into the original condition. In the fence example, 2(10 + 10) = 40, which matches.
- Compare with nearby values. Try x = 9 and x = 11. Both give an area of 99 m2, which is less than 100 m2. This supports a maximum.
- Check sense. Lengths, areas and volumes must be positive. If the answer is not sensible, the setup is wrong.
The comparison check is useful when the second derivative becomes long. It is a good habit, but in the board answer the second derivative test should still be written.
Also check the shape of the answer. For a fixed perimeter the best rectangle is a square. If a problem about a box or a cylinder gives a strange ratio, go back and recheck the condition used in Step 3.
How a tutor helps and a weekly routine
A tutor does not need to teach new calculus here. The tutor watches how your child reads the question and where the setup goes wrong. That is hard to see in a large class and easy to see one-to-one.
Great Home Tutors offers one-to-one tuition at your home. You tell us the class, subject, board and your locality. A person from our team then calls you back to understand the student. The first class is to see the fit before anything is fixed. Tutors we have checked ourselves carry a Verified mark.
Whatever support you choose, this routine works at home:
- Two problems a day, three days a week. Choose different shapes: a box, a cylinder, a cone, a wire, a fence.
- Setup first. Ask your child to write only Steps 1 to 3 and explain them aloud. Then differentiate.
- Weekly review. Keep a notebook of mistakes with the reason for each. Revise it before attempting new problems.
- Use NCERT first. Finish the NCERT examples and exercise of the chapter before adding other practice.
Key takeaways
- Most lost marks come from setting up, not from differentiating.
- Use the same six steps in the same order for every problem.
- Reduce to one variable before differentiating.
- Always test with the second derivative and write "maximum" or "minimum".
- Check by substituting back and comparing nearby values.
Frequently asked questions
Is the second derivative test compulsory in the board answer?
You need some test to show that the point is a maximum or a minimum. The second derivative test is the quickest and the one students usually write. Write the sign of the second derivative clearly.
What if the second derivative is zero at the critical point?
Then that test does not decide. Use the first derivative test instead. Check whether the sign of the first derivative changes on either side of the point.
How many problems should my child solve in a week?
Six well-checked problems, as in the routine above, are better than many rushed ones. Quality of setup matters more than count.
Can my child learn this without a tutor?
Yes, if the method is followed and mistakes are logged. A tutor helps when the same setup errors keep repeating and nobody can see why.
How do I ask for a home tutor in Dwarka or Gurugram?
Tell Great Home Tutors the class, subject, board and your sector or PIN. A person from the team calls you back to understand the student before a tutor is suggested.
What to do next. Try it yourself on the page below.
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