Linear algebraic solution of basic economic mathematics

First, if the tangent point is (a, y(a)) and y' =-2x, then the slope k=-2a.

Then the tangent equation is Y-y(a)=-2a(X-a)☆ where y(a)= 1-aa,

Find the intersection of this tangent with the X axis and the Y axis, assuming x0 and y0 respectively.

Then the area S= the area of the triangle x0*y0/2 -∫(0 to 1) 1-xxdx★

The integral in the above formula is a constant value =2/3, and we only need to find the best area of the triangle.

Or,

Area S(a)=∫(0 to a) The formula of tangent Y-the formula of parabola Y is 1-xxdx.

+∫(0 to y(a)) The formula of tangent X-the formula of parabola X is √ 1-ydy★

Find the minimum value of ★★.

It can be found that ★ = ★ = (1+AA) 2/4a-2/3, x0=( 1+aa)/2a, y0= 1+aa,

Find a =1√ 3, and the minimum area S(a)=4√3 /9-2/3.

Substituting a= 1/√3 into ☆ is the tangent equation.