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I recently finished a homework assignment with the exceptions of the following:

1.) f(x) =x^3 - x^2 + x, show there is a number c such that f(c)=10.

f(x) can be equated to 10, but I'm not quite sure how to solve the equation from that point.

2.) Prove that the equation has at least one real root.

e^x = 2 - x

In order to understand this question, i attempted to carry the same procedure with another equation: y = x^2 + x + 2. If the discriminant is 0, then there is a single root. If the discriminant is <0, no roots, and >0, multiple roots. But the same procedure doesn't work with the above equation, or for cubics, quartics, etc.

3.) For what values of x is F continuous?

f(x) = [ 0 if x is rational, 1 if x is irrational

I understand that the function can never be continuous, since it oscillates between 0 and 1 infinitely. But can anyone clarify what the following text means:

http://mathworld.wolfram.com/images/equations/DirichletFunction/equation3.gif [Broken]

1.) f(x) =x^3 - x^2 + x, show there is a number c such that f(c)=10.

f(x) can be equated to 10, but I'm not quite sure how to solve the equation from that point.

2.) Prove that the equation has at least one real root.

e^x = 2 - x

In order to understand this question, i attempted to carry the same procedure with another equation: y = x^2 + x + 2. If the discriminant is 0, then there is a single root. If the discriminant is <0, no roots, and >0, multiple roots. But the same procedure doesn't work with the above equation, or for cubics, quartics, etc.

3.) For what values of x is F continuous?

f(x) = [ 0 if x is rational, 1 if x is irrational

I understand that the function can never be continuous, since it oscillates between 0 and 1 infinitely. But can anyone clarify what the following text means:

http://mathworld.wolfram.com/images/equations/DirichletFunction/equation3.gif [Broken]

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