Question

if \(\large\tt \begin{align} \color{black}{f(x)=x^3-4x+p}\end{align}\)
and \(\large\tt \begin{align} \color{black}{f(0)}\end{align}\) and \(\large\tt \begin{align} \color{black}{f(1)}\end{align}\)
are of oppsoite signs ,then which of this is necessarily true?

Answer 1

\((a.)-1<p<2<-----\\
(b.)0<p<3\\
(c.)-2<p<1\\
(d.)-3<p<0\)

Answer 2

@kirbykirby

Answer 3

@aum , @freckles ,help me friends

Answer 4

\[
f(x) = x^3 - 4x + p \\
f(0) = 0 - 0 + p = p \\
f(1) = 1 - 4 + p = p - 3 \\
\text{f(0) and f(1) are of opposite signs. Two possibilities:} \\
p \lt 0 ~~\text{ and }~~p-3 > 0 ~~\text{ or } \\
p \gt 0 ~~\text{ and }~~p-3 < 0 \\
\text{First case is not possible. If p < 0, then p - 3 will also be less than 0.} \\
p \gt 0 ~~\text{ and }~~p-3 < 0 \implies 0 \lt p \lt 3
\]

Answer 5

thnk u very much @aum