if a function \(f(x)= 5x³-15x-a\) has three distinct and real roots for \(a € ( \alpha,\beta)\) then find \(\beta-2(\alp…

\(\begin{aligned}
f(x) & =5 x^3-15 x-a \\
f^{\prime}(x) & =15 x^2-15 \\
f^{\prime}(x) & =15\left(x^2-1\right) \\
& =15(x-1)(x+1)
\end{aligned}\)

\((-1) \rightarrow\) point of maxima \(1 \rightarrow\) point of minima

\(
\begin{aligned}
& f(1)=-10-9 \\
& f(-1)=10-9
\end{aligned}
\)

For 3 distinet meakerw)

\(
\begin{array}{lll}
-10-a<0 \Rightarrow & a>-10 \\
a \text { ard } \quad 10 a>0 & a<10
\end{array}
\)

\(\begin{aligned}
& =1 \quad a \in(-10,10) \\
& \alpha=-10, \quad \beta=10 \\
& \Rightarrow \beta-2 \alpha=10-2 \times(-10) \\
& =10+20 \\
& =30
\end{aligned}\)