examday

/TMUA

For each positive integer $n$, consider the function

Which of the following statements is/are necessarily true?

• I. If $n$ is even then $\displaystyle\int_{-2}^2 f_n(x)\,dx\leq34$.
• II. For every positive real number $k$, there exists a positive integer $n$ such that $f_n'(x)>k$ for some real number $x$ with $-1<x<0$.
• III. If $n>1$ then there exist real numbers $a$ and $b$ such that the equation $f_n(x)-ax=b$ has exactly four solutions.