Question

Maximum value of g(x) in x $ϵ$ [0,7] is.

• A. 3
• B. 9/2
• C. 3/2
• D. 6

Answer 1

The correct option is B 9/2

$g\left(x\right)={\int }_0^{x}f\left(t\right)dt$
$g^{\prime }\left(x\right)=f\left(x\right)$ From the graph it is clear that
$f\left(x\right)>0$ in $xϵ[0,3]$ and $f\left(x\right)<0$ in $xϵ[3,7]$
$g\left(x\right)$ is increasing in $[0,3]$ and $g\left(x\right)$ is decreasing in $[3,7]$
maximum value of $g\left(x\right)$ occurs at $x=3$
$g\left(3\right)={\int }_0^{3}f\left(t\right)dt$
$={\int }_0^{1}1\cdot dt+{\int }_1^2(2t-1)dt+{\int }_2^3(9-3t)dt$
$=1+{[t^2-t]}_1^2+{[9t-3\frac{t^2}{2}]}_2^3$
$=1+(4-2-0)+(27-\frac{27}{2}-18+6)=\frac{9}{2}$