Let f(x) = ∫ t(t^2-3t+20)dtx ∈ to (0, x), x ∈ to (1, 3) and range of f(x) is (α, β), then α+β is equal to

Question

Let \(f(x)=\int_0^x t\left(t^2-3 t+20\right) d t, x \in(1,3)\) and range of f(x) is \((\alpha, \beta),\) then \(\alpha+\beta\) is equal to

(1) \(\frac{185}{4}\)

(2) \(\frac{185}{2}\)

(3) \(\frac{185}{3}\)

(4) \(\frac{37}{4}\)

Answer 1

Correct option is (2) \(\frac{185}{2}\)   

\(f(x)=\frac{x^4}{4}-x^3+10 x^2\)   

\(f(x) =x^3-3 x^2+20 x \)

\( =x\left(x^2-3 x+20\right)\)

in \((1,3) f^{\prime}(x)\) is positive

\(\therefore f(x)\) is increasing in (1, 3)

\(\therefore \alpha=f(1)=\frac{37}{4}\)

\(\beta=f(3)=83.25=\frac{333}{4}\)

\(\therefore \alpha+\beta=92.5=\frac{185}{2}\)