solving an equation with terms of trig. and algebraic functions

Is there a general approach for solving functions that contain both trig. and algebraic terms? And to be more specific, how does it apply for a problem like this:

$\displaystyle f(x)=x^3+2x+tanx$

In problems with trig. functions, I'm able to determine that for $\displaystyle sin(x)=cos(x)$, $\displaystyle x$ is equal to $\displaystyle \frac{\pi}{4}, \frac{5\pi}{4}$ because it corresponds to an isosceles triangle on the unit circle with equal sides of lengths $\displaystyle cos(x)$ and $\displaystyle sin(x)$, but it wasn't a direct proof.

Any thoughts?

Re: solving an equation with terms of trig. and algebraic functions

Quote:

Originally Posted by

**Lambin** Is there a general approach for solving functions that contain both trig. and algebraic terms? And to be more specific, how does it apply for a problem like this:

$\displaystyle f(x)=x^3+2x+tanx$

In problems with trig. functions, I'm able to determine that for $\displaystyle sin(x)=cos(x)$, $\displaystyle x$ is equal to $\displaystyle \frac{\pi}{4}, \frac{5\pi}{4}$ because it corresponds to an isosceles triangle on the unit circle with equal sides of lengths $\displaystyle cos(x)$ and $\displaystyle sin(x)$, but it wasn't a direct proof.

I have several thoughts about all this.

One is that $\displaystyle f(x)=x^3+2x+\tan(x)$ is a function. **It is not a problem.**

So you need to post real problems. That is, actual questions.

Re: solving an equation with terms of trig. and algebraic functions

Okay. This actually stems from a calculus problem, but I figured I would need to solve it to determine critical points. The reason why I felt that a solution or solutions may exist was because I could imagine that for $\displaystyle -(x^3+2x)=tan(x)$, the two functions would intersect when drawn on a graph.

**QUESTION:**

Does $\displaystyle f(x)=x^3+2x+tanx$ have any local maximum or minimum values? Give reasons for your answer.

EDIT: Actually, I forgot to take the derivative and then solve the equation. But my question still stands. Also, let me know if I should delete this post, and repost on Calculus.

Re: solving an equation with terms of trig. and algebraic functions

Quote:

Originally Posted by

**Lambin** Okay. This actually stems from a calculus problem, but I figured I would need to solve it to determine critical points. The reason why I felt that a solution or solutions may exist was because I could imagine that for $\displaystyle -(x^3+2x)=tan(x)$, the two functions would intersect when drawn on a graph. **QUESTION:**

Does $\displaystyle f(x)=x^3+2x+tanx$ have any local maximum or minimum values? Give reasons for your answer.

Well, **very good**.

Now you need to solve $\displaystyle 3x^2+2+\sec^2(x)=0$. That is a hell of a difficult task.

I would suggest a graphical solution.