Question

The curve y = ax^2 + bx + c touches the line y = 2x at the origin and has a maximum point at x = 1. Find the values of a, b and c.

(answer: a = -1, b = 2, c = 0)

Answer 1

you have given equations you can derive..
touching at the origin: 0=0+0+c
max at x=1:y' = 0 =2a+b
max at x=1:y"= negative=2 *da/dx which means a has to be negative for the derivative to be negative.
touches y=2x at origin: Slope is 2, or y'=2ax+b so at origin, b=2
if b=2, then a=-1

c=0 b=-2a a=negative something
last clue: touchs the line at orgin which makes slope same as y=2x, and the slope is 2, so
y'=2ax+b=2 at x=0, so b=2
which finally leads to a: 0=-2a+b so a=