If a number is drawn at random from the set $\{1,3,5,7$, $\ldots, 59\}$, then the probability that it lies…
If a number is drawn at random from the set $\{1,3,5,7$, $\ldots, 59\}$, then the probability that it lies in the interval in which the function $f(x)=x^3-16 x^2+20 x-5$ is strictly decreasing, is
$\frac{1}{5}$
$\frac{1}{3}$
$\frac{1}{2}$
$\frac{1}{6}$
Solution
$\begin{aligned} & \text { } f(x)=x^3-16 x^2+20 x-5 \\ & f^{\prime}(x)=3 x^2-32 x+20=(3 x-2)(x-10) \\ & f^{\prime}(x) \lt 0 \\ & x \in\left(\frac{2}{3}, 0\right) \Rightarrow x \text { can take values } 1,3,5,7,9 \\ & n(\mathrm{~S})=30, n(\mathrm{E})=5 \\ & \text { Required probability }=\frac{n(\mathrm{E})}{n(\mathrm{~S})}=\frac{5}{30}=\frac{1}{6}\end{aligned}$