We have,
$$
\begin{aligned}
f(x) & =x^5-5 x^4+5 x^3-10 \\
f^{\prime}(x) & =5 x^4-20 x^3+15 x^2
\end{aligned}
$$
For maxima or minima put $f^{\prime}(x)=0$
$$
\begin{aligned}
\therefore \quad 5 x^4-20 x^3+15 x^2 & =0 \\
\Rightarrow \quad 5 x^2\left(x^2-4 x+3\right) & =0 \\
\Rightarrow \quad 5 x^2(x-3)(x-1) & =0 \\
x & =0,1,3 \\
f^{\prime \prime}(x) & =20 x^3-60 x^2+30 x \\
f^{\prime \prime}(x) & =10 x\left(2 x^2-6 x+3\right) \\
f^{\prime \prime}(0) & =0 \\
f^{\prime \prime}(1) & =10(2-6+3) < 0 \\
f^{\prime \prime}(3) & =30(18-18+3)>0
\end{aligned}
$$
$\therefore \quad$ Local maxima at $x=1$ and local minima at $x=3$ Here, $a=1, b=3$
$$
\therefore \quad 2 a+b=2+3=5
$$