Thank you for reporting, we will resolve it shortly
Q.
Number of integral values of $p$ for which the cubic $2 x^3-3 x^2+p=0$ has 3 real roots (not necessarily distinct), is
Application of Derivatives
Solution:
Draw graph of $y=p$ and $y=3 x^2-2 x^3$
Hence, for three roots $p \in[0,1]$. Ans.
Aliter: Let
$f(x)=2 x^3-3 x^2+p $
$f^{\prime}(x)=6 x(x-1)$
Now, $f (0) \cdot f (1) \leq 0 \Rightarrow p( p -1) \leq 0 \Rightarrow p \in[0,1]$.
