10. Given the function: $$f ( x ) = \left| 4 - x ^ { 2 } \right|$$ verify that it does not satisfy all the hypotheses of Rolle's theorem in the interval $[-3; 3]$ and that nevertheless there exists at least one point in the interval $[-3; 3]$ where the first derivative of $f(x)$ vanishes. Does this example contradict Rolle's theorem? Justify your answer thoroughly. \footnotetext{Maximum duration of the examination: 6 hours. The use of scientific and/or graphing calculators is permitted provided they are not equipped with symbolic computation capability (O.M. no. 257 Art. 18 paragraph 8). The use of a bilingual dictionary (Italian – language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the dictation of the theme. }
10. Given the function:
$$f ( x ) = \left| 4 - x ^ { 2 } \right|$$
verify that it does not satisfy all the hypotheses of Rolle's theorem in the interval $[-3; 3]$ and that nevertheless there exists at least one point in the interval $[-3; 3]$ where the first derivative of $f(x)$ vanishes. Does this example contradict Rolle's theorem? Justify your answer thoroughly.
\footnotetext{Maximum duration of the examination: 6 hours.\\
The use of scientific and/or graphing calculators is permitted provided they are not equipped with symbolic computation capability (O.M. no. 257 Art. 18 paragraph 8).\\
The use of a bilingual dictionary (Italian – language of the country of origin) is permitted for candidates whose native language is not Italian.\\
It is not permitted to leave the Institute before 3 hours have elapsed from the dictation of the theme.
}