Question: The function \(f\) has a derivative everywhere and has just one critical point, at \(x=3\). In parts (a)-(d), you are given additional conditions. In each case decide whether \(x=3\) is a local maximum, a local minimum, or neither. Explain your reasoning. Sketch possible graphs for all four cases.

1. \(f^{\prime}(1)=3\) and \(f^{\prime}(5)=-1\)
2. \(f(x) \rightarrow \infty\) as \(x \rightarrow \infty\) and as \(x \rightarrow-\infty\)
3. \(f(1)=1, f(2)=2, f(4)=4, f(5)=5\)
4. \(f^{\prime}(2)=-1, f(3)=1, f(x) \rightarrow 3\) as \(x \rightarrow \infty\)

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