# Find the maximum length of line of ST and At what value of x will ST be at it's longest?

Given: The sketch shows the graph of a function f, which is a straight line defined by y=mx+k such that the point V lies on the line and the graph of a function g, which is a parabola with vertex R. The straight line and parabola intersect at P and . The points S and T lie on the parabola and straight line, respectively, between P and Q. The line ST is parallel to the y-axis.

V(-3,4) R(3,2) Point where parabola intersects the x-axis (1;0)

I have worked my way through the questions and have found: Q(5,0) P(2,3/2) Straight line: y=-1/2x+5/2 Parabola: g(x)=-1/2x^2+3x-5/2

How would i go about finding the co ordinates of ST? Do i assume that T is the midpoint of P and Q?

You have to maximize the difference of $g(x)$ and $y(x)$. This can be done by differentiation.
$d(x)=g(x)-y(x)=-\frac12\cdot x^2+3\cdot x-\frac52 -( -\frac12\cdot x+5/2)$
$=-\frac12\cdot x^2+3.5\cdot x-5$
$d'(x)=-x+3.5$
Now calculate $x_0$, where $d'(x_0)=0$
If $d''(x_0)<0$, then $x_0$ is the x-value of the maximum of $d(x)$