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UPSC 2018 Maths Optional Paper 1 Q4a — Step-by-Step Solution

13 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 values of $x^4-5x^2+4$ on the interval $[2,3]$.

Technique

Closed-interval extrema = compare critical values inside $[a,b]$ with endpoint values; here no critical point is interior, so endpoints decide.

Solution

Step 1 — Critical points

Let $f(x)=x^4-5x^2+4$. Then

$f'(x)=0$ at $x=0$ and $x=\pm\sqrt{5/2}=\pm\sqrt{2.5}\approx\pm1.581$. None of these lies in $[2,3]$.

Step 2 — Monotonicity on $[2,3]$

For $x\in[2,3]$: $2x>0$ and $2x^2-5\ge 2(4)-5=3>0$, so $f'(x)>0$ throughout. Thus $f$ is strictly increasing on $[2,3]$, and the extrema occur at the endpoints: minimum at $x=2$, maximum at $x=3$.

Step 3 — Evaluate at the endpoints

Answer