If it is not Even

Algebra Level 3

Consider $f\left( x \right) =\ln { { \left( \dfrac { { 2x }^{ 3 }+1 }{ 1-{ 2x }^{ 3 } } \right) }^{ 3 }+\dfrac { 1 }{ 8 } \sin { x } }$ . Find the sum of all the roots of $f(x)$ .

The answer is 0.

1 solution

Otto Bretscher
Apr 20, 2016

We can spot the root $x=\boxed{0}$ . Since the function is increasing in its domain, $|x|<\sqrt[3]\frac{1}{2}$ , this is the only root.

Correct. But even though we have absolutely no knowledge about roots of f(x) we realize that f(x) is the sum of two odd functions (which is an odd function again) whereat the roots are mirrored. That's why their sum must be necessarily equal 0.

Andreas Wendler - 5 years, 1 month ago

True! I actually find it (even) easier to see that this function is increasing than to see that it is odd. Also, I wanted to give a different solution for the sake of variety... I have used this even/odd argument a few times recently.

Otto Bretscher - 5 years, 1 month ago

If it is not Even

The answer is 0.

1 solution

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