Find the equation of a quartic polynomial whose graph is symmetric about the y-axis and has local maxima at (−3,3) and (3,3) and a y-intercept of 2.

Answer 1

you know that y' = ax(x-3)(x+3) = a(x^3-9x)
That makes
y = a(1/4 x^4 - 9/2 x^2) + C
Since y(0) = 2, C=2 and you have
y = ax^2(1/4 x^2 - 9/2) + 2
Since y(3)=3, you have
9a(9/4 * 9 - 9/2) + 2 = 3
a = -4/81
and so you have
y = -4/81 x^2 (1/4 x^2 - 9/2) + 2

see the graph and its properties at

https://www.wolframalpha.com/input/?i=-4%2F81+x%5E2+%281%2F4+x%5E2+-+9%2F2%29+%2B+2

Answer 2

thank you so much! :)