Probability that X is greater than the mode of X? How do I solve this problem:
Let X be a continuous random variable with density function
\begin{equation}
f(x) =
  \begin{cases} 
      \hfill ax^2e^{-10x}    \hfill & \text{for x $\geq$ 0} \\
      \hfill 0 \hfill & \text{otherwise} \\
  \end{cases}
\end{equation}
where a > 0. What is the probability of X greater than or equal to the mode of X?


What is the probability of $X$ greater than or equal to the mode of $X$?

Hint. One may find the maximum of $f$ over $\mathbb{R}$, using
\begin{equation}
f'(x) =
  \begin{cases} 
      \hfill -2ax(5x-1)e^{-10x}    \hfill & \text{for $x \geq$ 0} \\\\
      \hfill 0. \hfill & \text{otherwise} \\
  \end{cases}
\end{equation}
Then one may evaluate
$$
P\left(X\geq \frac15\right)=\int_{1/5}^\infty-2ax(5x-1)e^{-10x}\:dx.
$$ The latter integral may be computed by parts.
