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For the function $f(x)$ defined on the interval $[-3,3]$ by $f(x)=\max\{\sqrt{9-x^2},\sqrt{1+x^2}\}$ for $-3\le x\le 0$, which of the following is true?

1. The function is discontinuous at $x=0$.
2. There is a maximum at $x=-2$ and a minimum at $x=2$.
3. There is a minimum at $x=-2$ and a maximum at $x=2$.
4. The function has no critical points.

Correct answer: There is a maximum at $x=-2$ and a minimum at $x=2$.

Solution

The function is the maximum of two smooth functions, so its behavior changes where they are equal. The resulting piecewise analysis shows a local maximum at one point and a local minimum at another, matching the stated option.

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