Question

What is the difference between the greatest and least values of the function

$\varphi \left(x\right)={\int }_0^x(t+1)dt$ on $[2,3]$

• A. $3$
• B. $2$
• C. $7/2$
• D. $11/2$

Answer 1

The correct option is C $7/2$

$\varphi \left(x\right)={\int }_0^x(t+1)dt={[\frac{t^2}{2}+t]}_0^x=\frac{x^2}{2}+x$
For $2<x<3$ we get $4<\frac{x^2}{2}+x<\frac{15}{2}$
Therefore difference between greatest value and least value is
$\frac{15}{2}-4=\frac{7}{2}$