The general fourth degree polynomial: \(\displaystyle{p}{\left({x}\right)}={a}{x}^{{4}}+{b}{x}^{{3}}+{c}{x}^{{2}}+{\left.{d}{x}\right.}+{e}\)

Since the graph is symmetric about y-axis, terms with odd powers of x must be zero⇒b=d=0. The y-intercept: \(\displaystyle{y}{\left({0}\right)}={e}={0}⇒{e}={0}\)

Reduced polynomial so far: \(\displaystyle{p}{\left({x}\right)}={a}{x}^{{4}}+{c}{x}^{{2}}\).

Critical points: \(p'(x)=4ax^3+2cx ⇒p'(x)=2x(2ax^2+c)\)

Critical points (except x=0): \(\displaystyle{x}=\pm\sqrt{{-\frac{{c}}{{2}}{a}}}\) We have these two equations: \(\displaystyle{1}=\sqrt{{-\frac{{c}}{{2}}{a}}}\)

a+c=2

Solving these gives a=-2 and c=4. Resulting polynomial: P(particular)=\(\displaystyle-{2}{x}^{{4}}+{4}{x}^{{2}}\)