Question 1142316
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If a function *[tex \Large f] is continuous on the closed interval *[tex \Large \[a,b\]] and has values *[tex \Large f(a)] and *[tex \Large f(b)] at each end of the domain interval, then the function takes on any value in the interval *[tex \Large \[f(a),f(b)\]] at some point in the domain interval.


All polynomial functions with real coefficients are everywhere continuous.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ x^7\ +\ 4x^5\ +\ 3x\ +\ 5]


is a polynomial function with real coefficients and is therefore everywhere continuous and is therefore continuous on the interval *[tex \Large \[-1,0\]].


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(-1)\ =\ -3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(0)\ =\ 5]


Therefore *[tex \Large f(x)] has any value in the interval *[tex \Large \[-3,5\]] for some *[tex \Large x\ \in\ \[-1,0\]]


*[tex \Large 0\ \in\ \[-3,5\]] therefore *[tex \Large \exists\,x_i\ \in\ \[-1,0\]\ :\ f(x_i)\ =\ 0]
								
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
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