Prove that there is no line through (-1,2) that is tangent to the graph of f (x) = x^2+ 3x .
The line through (-1, 2) : y = mx + c
Subs. (-1, 2) to the equation gives c = 2 + m, so the equation of the line : y = mx + 2 + m
If the line is tanget to the curve, then they will have common point, so :
mx + 2 + m = x^2 + 3x, where m = f '(x) = 2x + 3