# Challenge Problems

## 18 thoughts on “Challenge Problems”

1. Aori Nevo says:

Prove or Disprove the following statement:
$$\int 2 \sin x \cos x \; dx= -\frac{\cos{2x}}{2}+C_1$$
and
$$\int 2 \sin x \cos x \; dx= \sin^2{x} + C_2$$

2. Aori Nevo says:

Let $$f(x)= \sin x$$. Does there exist an equation of a parabola: $$g(x) = ax^2 + bx + c$$ ($$a \not = 0$$) such that for at least three values of $$x$$, the function $$f$$ and $$g$$ agree?

3. Aori Nevo says:

Prove/Disprove: If $$y = b$$ is a horizontal asymptote then $$f(x) \not = b$$ for any $$x$$.

4. Aori Nevo says:

Does there exist a function which is both even and odd? If so, find all such functions.

1. Anonymous says:

This is probably not the solution you are looking for but, f(x) = 0 is both even and odd.

2. Is this the only one? If so, prove that there are no other functions that are both even and odd.

5. Aori Nevo says:

Prove/Disprove: If $$f'(x) >0$$ for $$x < a$$  and $$f'(x) < 0$$ for $$x > a$$, the $$x = a$$ is a relative maximum.

6. Aori Nevo says:

Prove/Disprove: If $$x = a$$ is a vertical asymptote of $$f(x)$$, then $$f(x)$$ is undefined at $$x = a$$.