examday

/TMUA

A function $f(x)$ of real numbers is called even if and only if $f(x)=f(-x)$ for all $x$ in the domain of $f(x)$. It is called odd if and only if $f(x)=-f(-x)$ for all $x$ in the domain.

For instance:

• $f(x)=|x|$ for all $x\in\R$ is even
• $f(x)=x$ for all $x\in\R$ is odd
• $f(x)=\sqrt{x^2-4}$ for $x$ with $|x|\ge2$ is even
• $f(x)=\sqrt{x-2}$ for $x$ with $x\ge2$ is neither even nor odd

Which of the following statements must be true?

• 1. If $f(x)$ is odd, then $\dfrac{f'(x)}{f'(-x)}=-1$ for all $x$ in the domain
• 2. If $f(x)$ is even, then $\dfrac{f'(x)}{f'(-x)}=-1$ for all $x$ in the domain
• 3. If $f(x)$ is odd, then $f(0)=0$
• 4. If $f(x)$ is even, then $f(0)=0$
• 5. If $f(x)$ is odd, then $\displaystyle\int_{-a}^af(x)\dd x=0$ for all $a$ in the domain
• 6. If $f(x)$ is even, then $\displaystyle\int_{-a}^af(x)\dd x=0$ for all $a$ in the domain