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)

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

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