Chapter 4.1, Problem 86E

(a)

To determine

To find: The slope of the line tangent to the parabola at A and find the slope of the line that is perpendicular to the tangent line at A, where position of A is $\left(a,a^2\right)$, $a>0$.

(b)

To determine

To find: The equation of the line joining A and B, when A is at $\left(a,a^2\right)$.

(c)

To determine

To find: The position of B on the parabola when A is at $\left(a,a^2\right)$.

(d)

To determine

The function $F\left(a\right)$ that gives the square of the distance between A and B.

(e)

To determine

To find: The critical point of $F\left(a\right)=\frac{{\left(4a^2+1\right)}^3}{16a^4}$.

(f)

To determine

To evaluate: The value of F at the critical point, the positions of A and B that minimize the length of L and the minimum length.

(g)

To determine

To draw: The graph of $F\left(a\right)=\frac{{\left(4a^2+1\right)}^3}{16a^4}$.