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UPSC 2019 Maths Optional Paper 1 Q3a — Step-by-Step Solution

15 marks · Section A

Maxima and minima of single-variable functions · Calculus · asked 7× in 13 yrs · Read the full method →

Question

Find the maximum and the minimum value of the function $f(x)=2x^3-9x^2+12x+6$ on the interval $[2,3]$.

Technique

Closed-interval extremum: critical points in the interval + endpoints; here $f$ is monotone on $[2,3]$ so endpoints decide.

Solution

Step 1 — Critical points

$f'(x)=0$ at $x=1$ and $x=2$. Of these, only $x=2$ lies in $[2,3]$ (it is the left endpoint).

Step 2 — Monotonicity on $[2,3]$

For $x\in(2,3)$: $(x-1)>0$ and $(x-2)>0$, so $f'(x)=6(x-1)(x-2)>0$. Thus $f$ is strictly increasing on $[2,3]$. On a closed interval where $f$ is monotone increasing, the minimum is at the left endpoint and the maximum at the right endpoint.

Step 3 — Evaluate at the endpoints

Answer