Math Help - solving an equation with terms of trig. and algebraic functions

1. 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:

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

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

Any thoughts?

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

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:
$f(x)=x^3+2x+tanx$
In problems with trig. functions, I'm able to determine that for $sin(x)=cos(x)$, $x$ is equal to $\frac{\pi}{4}, \frac{5\pi}{4}$ because it corresponds to an isosceles triangle on the unit circle with equal sides of lengths $cos(x)$ and $sin(x)$, but it wasn't a direct proof.
I have several thoughts about all this.

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

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

3. 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 $-(x^3+2x)=tan(x)$, the two functions would intersect when drawn on a graph.

QUESTION:

Does $f(x)=x^3+2x+tanx$ 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.

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

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 $-(x^3+2x)=tan(x)$, the two functions would intersect when drawn on a graph. QUESTION:
Does $f(x)=x^3+2x+tanx$ have any local maximum or minimum values? Give reasons for your answer.
Well, very good.
Now you need to solve $3x^2+2+\sec^2(x)=0$. That is a hell of a difficult task.
I would suggest a graphical solution.