GATE 2022 BT – Question 32
The maximum of $f(x)=3x^2-x^3$ for $x>0$ is ___________.
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Correct answer: 4
Explanation
**Step 1: stationary points.**
$$f(x)=3x^2-x^3,\qquad f^\prime(x)=6x-3x^2=3x(2-x).$$
$f^\prime=0$ at $x=0$ or $x=2$. For $x>0$ the candidate is $x=2$.
**Step 2: nature.** $f^{\prime\prime}(x)=6-6x$, and $f^{\prime\prime}(2)=-6<0$, so $x=2$ is a **maximum**.
**Step 3: value.**
$$f(2)=3(4)-8=12-8=\mathbf{4}.$$
(For large $x$ the cubic term makes $f$ negative, so this local maximum is the maximum for $x>0$.)