If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then

If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then
  1. $\mathrm{f}(x)$ is decreasing in $(-\infty, 10]$ and increasing in $[10, \infty)$
  2. $\mathrm{f}(x)$ is increasing in $(-\infty, 10]$ and decreasing in $[10, \infty)$
  3. $\mathrm{f}(x)$ is increasing throughout real line
  4. $\mathrm{f}(x)$ is decreasing throughout real line

Solution

$\begin{aligned} & \mathrm{f}(x)=x^3-10 x^2+200 x-10 \\ & \Rightarrow \mathrm{f}^{\prime}(x)=3 x^2-20 x+200 \end{aligned}$
For $\mathrm{f}(x)$ to be increasing $\mathrm{f}^{\prime}(x)\gt0$ $\begin{aligned} & \Rightarrow 3 x^2-20 x+200\gt0 \\ & \Rightarrow 3\left(x^2-\frac{20}{3} x+\frac{200}{3}+\frac{100}{9}-\frac{100}{9}\right)\gt0 \\ & \Rightarrow 3\left[\left(x-\frac{10}{3}\right)^2+\frac{500}{9}\right]\gt0 \\ & \Rightarrow 3\left(x-\frac{10}{3}\right)^2+\frac{500}{3}\gt0 \end{aligned}$
Always increasing throughout real line.

Asked in: MHT CET 2024 (15 May Shift 1)

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