Mathematics - Definite Integration Question with Solution | TestHub

$f^{\prime }(x)=x\left(x^2-9x+20\right),x\in (1,5)$
$=(x-4)x(x-5)$

$\begin{array}{rl}& \Rightarrow f^{\prime }(x)>0\forall x\in (1,4)\\ & \Rightarrow f^{\prime }(x)<0\forall x\in (4,5)\\ & \Rightarrow f(x)\text{ increasing in }(1,4)\\ & f(x)\text{ decreasing in }(4,5)\\ & \Rightarrow \text{ critical points to check: }\end{array}$
$\begin{array}{rl}& x=1,4,5\\ & f(x)={\int }_0^x\left(t^3-9t^2+20t\right)dt\\ & =\frac{t^4}{4}-3t^3+{10t^2|}_0^x=\frac{x^4}{4}-3x^3+10x^2\\ & f(1)=\frac{1}{4}-3+10=\frac{29}{4}\\ & f(4)=4^3-3.4^3+10.4^2=-2.4^3+10.4^2=32\\ & f(5)=\frac{5^4}{4}-3.5^3+10.25=\frac{5^4}{4}-125=\frac{125}{4}\end{array}$
$\begin{array}{rl}& \text{ Range }\Rightarrow \left[\frac{29}{4},32\right]\Rightarrow 4(\alpha +\beta )=128+29\\ & =157\end{array}$