What is the smallest possible value of c where three distinct and real solutions are possible in the equation – e^x = cx^2

Dear student 

For k≤0, there will be no real solution at all for kx^2≤0
For k>0 there will always be exactly one solution with x≤0. For small positive k, there is no positive solution, for suitable k the curves will touch and for larger k we have two positive real solutions (so three real solutions in total). We ned to find the boundary case that the curves touch, i.e., find k>0 and x0>0 such that the functions and derivative agree at x

Good Luck