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

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The function f(x)=(x-a)^2*(x-b)^2+x is continuous as it is a polynomial. It is established that f(a)=a and f(b)=b, meaning the values at the endpoints are a and b. The value (a+b)/2 lies between a and b, regardless of their order. By the Intermediate Value Theorem, there exists some x in ℝ such that f(x)=(a+b)/2. Thus, the function indeed takes on the value (a+b)/2 for some x.
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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 $f(x)=\dfrac{a+b}{2}$.
 
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