Find k if the function f(x) is defined by
`f(x)=kx(1-x),"for "0ltxlt1`
=0 , otherwise, is the probability density function (p.d.f.) of a random varible (r.v) X. Also find P `(Xlt(1)/(2))`

Explanation

To find the value of \( k \) such that the function \( f(x) = kx(1-x) \) for \( 0 < x < 1 \) is a probability density function (p.d.f.), we need to ensure that the total area under the curve of \( f(x) \) over the interval from 0 to 1 equals 1.

### Step 1: Set up the integral
We need to compute the integral of \( f(x) \) from 0 to 1:
\[ \int_0^1 f(x) \, dx = \int_0^1 kx(1-x) \, dx \]
This integral must equal 1 for \( f(x) \) to be a valid p.d.f.
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