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Applications of Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives1 Mark
Question
Test whether the function $f(x)=x^3+6 x^2+12 x-5$ is increasing or decreasing for all $x \in \mathrm{R}$.
✓

Answer

Given that $f(x)=x^3+6 x^2+12 x-5$
Differentiate $w . r . t . x$.
$
\begin{aligned}
& f^{\prime}(x)=3 x^2+12 x+12=3\left(x^2+4 x+4\right) \\
& f^{\prime}(x)=3(x+2)^2
\end{aligned}
$
$3(x+2)^2$ is always positive for $x \neq-2$
$
\therefore f^{\prime}(x) \geq 0 \text { for all } x \in \mathrm{R}
$
Hence $f(x)$ is an increasing function for all $x \in \mathrm{R}$.

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