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Mathematics · 2024 · 2 marks
CBSE 2024 · Region 1 · Set 1 · Q22
Show that the function f ( x )=4 x^3-18 x^2+27 x -7 has neither maxima nor minima.
Show that the function $\displaystyle \mathrm{f}(\mathrm{x})=4 \mathrm{x}^{3}-18 \mathrm{x}^{2}+27 \mathrm{x}-7$ has neither maxima nor minima.
Marking-scheme solution
$\displaystyle \mathrm{f}^{\prime}(\mathrm{x})=12 \mathrm{x}^{2}-36 \mathrm{x}+27$
$\displaystyle =3(2 \times-3)^{2} \geq 0$ for all $\displaystyle \mathrm{x} \in R$
$\displaystyle \therefore \mathrm{f}$ is increasing on R .
Hence $\displaystyle \mathrm{f}(\mathrm{x})$ does not have maxima or minima.
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