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Karla's question at Yahoo! Answers (Intermediate Value Theorem).

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello Karla,

The function $f(x)=(x-a)^2(x-b)^2+x$ is continuos in $\mathbb{R}$ (polynomical function). Besides, $f(a)=a$ and $f(b)=b$.

But $\dfrac{a+b}{2}$ is the middle point of the segment with endpoints $a$ and $b$ (no matter if $a<b$, $b<a$ or $a=b$) so, $\dfrac{a+b}{2}$ is included between $a$ and $b$. According to the Intermediate Value Theorem, there exists $x\in\mathbb{R}$ such that $f(x)=\dfrac{a+b}{2}$.