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
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asked 2021-05-13
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).

Answers (1)

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

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