Calculus
posted by Lynnn .
Consider line segments which are tangent to a point on the right half (x>0) of the curve y = x^2+7 and connect the tangent point to the xaxis. If the tangent point is close to the yaxis, the line segment is long. If the tangent point is far from the yaxis, the line segment is also very long. Which tangent point has the shortest line segment?
(Suppose C is a positive number. What point on the curve has first coordinate equal to C?)

answer to C question at the end: If 1st coordinate (xvalue) is C, the 2nd coordinate (yvalue) is C^2+7.
Now, for the other question:
At (x,y), the slope of the tangent line is
dy/dx = 2x
So, now we have a point (a,a^2+7) and a slope (2a).
The equation of the tangent line where x=a is thus
y(a^2+7) = 2a (xa)
y = 2ax  2a^2 + a^2+7
y = 2ax  a^2 + 7
This line intersects the xaxis where
0 = 2ax  a^2 + 7
x = (a^27)/2a
So, the length of the line segment from (a,a^2+7) to (0,(a^27)/2a) is
d^2 = (a  (a^27)/2a)^2 + (a^2+7)^2
or
d = 1/2a (a^2 + 7) sqrt(4a^2 + 1)
dd/da = (8a^4 + a^2  7)/[2a^2 sqrt(4a^2+1)]
dd/da = 0 when x = ±√(7/8) = ±0.935
d = 8.929