Question

# Find a formula for the function described. A fourth-degree polynomial whose graph is symmetric about the y-axis, has a y-intercept of 0, and global ma

Polynomial graphs
Find a formula for the function described. A fourth-degree polynomial whose graph is symmetric about the y-axis, has a y-intercept of 0, and global maxima at (1, 2) and (-1, 2).

2021-05-14

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)=$$\displaystyle-{2}{x}^{{4}}+{4}{x}^{{2}}$$