If the area by `y=x^2+2x-3` and the line `y=k x+1` is the least, find `k` and also the least area.

Explanation

Solving we obtain
`x^(2)+(2-m) x-4=0`
Let `alpha, beta` be roots `rArr alpha + beta = m - 2 , alpha beta = -4`
`A(m) = |underset(alpha)overset(beta)int(,x+1-x^(2)-2x+3)dx|=|underset(alpha)overset(beta)(-x^(2)+(m-2)x+4)dx|`
`=|(-(x^(3))/(3)+(m-2)'(x^(2))/(2)+4x)_(alpha)^( beta)|=|(alpha^(3)-beta^(3))/(3)+(m-2)/(2)(beta^(2)-alpha^(2))+4(beta-alpha)|`
`=|beta-alpha|.|-1/3(beta^(2)+betaalpha+alpha^(2))+((m-2))/(2)(beta-alpha)+4|`
`= sqrt((m-2)^(2)+16)|-1/3((m-2)^(2)+4)+((m-2))/(2)(m-2)+4|=sqrt((m-2)^(2)+16)|1/6 (m-2)^(2)+8/3|`
`A(m)((m-2)^(2)+16)^(3//2) 1/6 rArr "Least" A(m) = 1/6 (16)^(3//2) = 32/3`.