Optimization Help

[frigid]

Can I get help with this optimization question?

Let T be the triangle with vertices at a= (-1,0), b= (1,0), and c= (0,2).

a) a point p lies inside T along the y-axis. Find the smallest possible value for the sum
lp-al + lp-bl + lp-cl (absolute value bars)(where lp-al is the distance between p and a)
b) A smaller triangle S, with its lower vertex (0,0) and its upper edge parallel to the x-axis, is inscribed in T. Determine the minimum possible perimeter of S.

 

[pappus]

Can I get help with this optimization question?

Let T be the triangle with vertices at a= (-1,0), b= (1,0), and c= (0,2).

a) a point p lies inside T along the y-axis. Find the smallest possible value for the sum
lp-al + lp-bl + lp-cl (absolute value bars)(where lp-al is the distance between p and a)
b) A smaller triangle S, with its lower vertex (0,0) and its upper edge parallel to the x-axis, is inscribed in T. Determine the minimum possible perimeter of S.

to a):

The point T has the coordinates \(\displaystyle \displaystyle{T(0, t), 0 \le t \le 2}\). Use Pythagorean theorem to determine the 3 distances. Use the 1st derivation of the distances to find an extremum. Proof that this extremum is a minimum.

to b):

Let h be the height of the smaller triangle. The line y = h intersects the slanted sides of the larger triangle in the points L and R. Determine the perimeter of the triangle. Use differentiation to find the minimum perimeter.