Calculus Applications: The Maxima-Minima Diagnostic
Every maxima-minima question carries a fixed quantity. Finding it is the entire problem; the differentiation after that is routine.
A topic note on how to recognise the geometry behind a JEE maxima-minima question and set up the single-variable function without inventing constraints that are not there.
How this actually works
Find the constraint, then the variable
Every maxima-minima question carries a fixed quantity — a sum, a product, a perimeter, a volume. That is the constraint, and it sets the domain of the single variable. The trap is to write a two-variable function and then invent an artificial relation between the variables, which is just the constraint hiding in a disguise. Read the constraint first, solve it for one variable, and substitute before differentiating.
Geometry before algebra
The second pattern is geometric — the area of a triangle cut from a corner, the volume of a cylinder inscribed in a sphere, the length of a ladder around a corner — where the variable is an angle or a position. In those, the cleanest move is to parametrise the geometry rather than to chase lengths. Let the angle be θ, write the area in terms of θ, and the algebra follows. JEE asks one of each type per January, and the geometric one rewards a diagram more than any amount of symbolic manipulation.
The second-derivative check, and the boundary it hides
A candidate who skips the boundary test loses marks on the endpoint that the stationary point does not see. The function is defined on a closed interval, so compare the stationary value against both boundary values. If a boundary value is smaller (for a minimum), the answer is not at the stationary point — which is the conclusion most students refuse to believe because they have already invested in the derivative.
The algebraic trap: division by zero
When you differentiate and set the result to zero, the algebra often produces a factor that was multiplied or divided through during setup. A candidate who divided by x during the substitution, and then solves differentiate = 0, will find x = 0 as a critical point — but x = 0 was excluded at the setup stage. Always carry the domain restriction forward into the derivative step.
Calculus Applications: The Maxima-Minima Diagnostic: the facts to keep
| Step one | Find the fixed quantity - a sum, product, perimeter or volume - because it sets the domain of the single variable |
|---|---|
| Step two | Solve the constraint for one variable and substitute before differentiating |
| Geometric variant | Parametrise by an angle or position rather than chasing lengths |
| The boundary test | On a closed interval, compare the stationary value against both endpoints |
| The algebraic trap | A factor you divided through during setup can reappear as a critical point after differentiation |
| Which test to use | Sign change of the first derivative - it does not lie when the second derivative is zero |
Questions students ask about this
Should I use the second derivative test or the first derivative test?
Use the sign change of the first derivative. It is faster, and it does not lie when the second derivative happens to be zero, which happens more often in JEE than students expect.
Do I need to check endpoints even for a continuous function?
Yes, always. A continuous function on a closed interval attains its maximum either at a critical point or at an endpoint. The question is which, and the only way to know is to check both.