Answer
maximum value: $5/4$ when $x=3/2$.
Work Step by Step
Let's compare $f(x)=-x^2 + 3 x-1$ to $f(x)=ax^2+bx+c$. We can see that $a=-1, b=3, c=-1$. $a\lt0$, hence the graph opens down, and hence its vertex is a maximum. The maximum value is at $x=-\frac{b}{2a}=-\frac{3}{2\cdot(-1)}=1.5.$ Thus, the maximum value is $f(1.5)=-(1.5)^2+3(1.5)-1=1.25.$
