See the coversheet for instructions and the point value of each problem.
- For each part below, sketch the graph of a function that
satisfies all the conditions or explain why it is impossible to
satisfy all the conditions. Conditions given in terms of \(x\)
(such as \(f(x) > 0\)) apply for all \(x\), whereas conditions
given in terms of \(a\) (such as \(f'(a) > 0\)) apply for a
single value \(x=a\), which you should label.
- \(f(x) > 0\), \(f'(x) > 0\), and \(f''(x) > 0\)
- \(f(x) < 0\), \(f'(x) > 0\), and \(f''(x) > 0\)
- \(f(x) > 0\), \(f'(x) < 0\), and \(f''(x) > 0\)
- \(f(x) > 0\), \(f'(x) > 0\), and \(f''(x) < 0\)
- \(f(x) > 0\), \(f'(x) < 0\), and \(f''(x) < 0\)
- \(f(x) < 0\), \(f'(x) > 0\), and \(f''(x) < 0\)
- \(f(x) < 0\), \(f'(x) < 0\), and \(f''(x) > 0\)
- \(f(x) < 0\), \(f'(x) < 0\), and \(f''(x) < 0\)
- \(f'(a) = 0\), and \(f''(x) > 0\)
- \(f'(a) = 0\), and \(f''(x) < 0\)
- \(f'(a) = 3\), \(f'(x) > 0\), and \(f''(x) = 0\)
- \(\displaystyle \lim_{x\rightarrow a^-}f(x)=-2\), \(\displaystyle \lim_{x\rightarrow a^+}f(x)=1\), \(f(a)=3\)
- \(\displaystyle \lim_{x\rightarrow a}f(x)=4\), \(f(a)=3\), \(f'(a)=0\)
- \(f(a)=1\), \(f(a+1)=2\), \(f'(x) > 0\)
- \(f(a)=1\), \(f(a-1)=2\), \(f'(x) > 0\)
- Find values for \(m\) and \(b\) so that
\(\displaystyle f(x)=
\begin{cases}
x^2 & \text{if \(x\le -2\)}\\
mx+b & \text{if \(x> -2\)}
\end{cases}\) is differentiable at \(x=-2\).
Last modified: Thu Sep 13 15:51:05 UTC 2018