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Mathematics · 2025 · 2 marks
CBSE 2025 · Region 4 · Set 2 · Q25
Find the local maxima and local minima of the function f(x)=8/3 x^3-12 x^2+18 x+5.
Find the local maxima and local minima of the function $\displaystyle f(x)=\frac{8}{3} x^{3}-12 x^{2}+18 x+5$.
Marking-scheme solution
\[\begin{aligned}
& f(x)=\frac{8}{3} x^{3}-12 x^{2}+18 x+5 \\
& \Rightarrow f^{\prime}(x)=8 x^{2}-24 x+18 \\
& \quad=2\left(4 x^{2}-12 x+9\right)=2(2 x-3)^{2}
\end{aligned}
\]
For critical points, Put $\displaystyle f(x)=0$
\[\Rightarrow 2(2 x-3)^{2}=0 \Rightarrow x=\frac{3}{2}
\]
since $\displaystyle f^{\prime}(x)$ does not change the sign as crosses $\displaystyle x=\frac{3}{2}$ from left to right, $\displaystyle f$ has no local maxima or local minima.
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CBSE Class 12 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.