PART 5：例題-最大最小【95中興行銷所】

設 \(f(x)=\sqrt{x}{{\left( x-4 \right)}^{{1}/{3}\;}}\) ，假設 \(c,d \in [0\;,\;6]\) ，

使 \(f(c) \le f(x) \le f(d)\;\;\forall x \in [0\;,\;6]\) ，則 \((c,d) = ?\)

找出所有的臨界點

SOL:  

\(f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{\left( {x - 4} \right)^{\frac{1}{3}}} + \frac{1}{3}{x^{\frac{1}{2}}}{\left( {x - 4} \right)^{ - \frac{2}{3}}}\) ，

\(\Rightarrow f'(x) = \frac{1}{6}{x^{ - \frac{1}{2}}}{\left( {x - 4} \right)^{ - \frac{2}{3}}}\left[ {3\left( {x - 4} \right) + 2x} \right]\)

\(\Rightarrow f'(x) = \frac{1}{6}{x^{ - \frac{1}{2}}}{\left( {x - 4} \right)^{ - \frac{2}{3}}}\left[ {5x - 12} \right]\)

\(x = 0\) 與 \(x = 4\) 與 \(x = \frac{2}{5}\) 為臨界值

比大小

\(f(0) = 0 \)

\(f(4) = 0 \)

\(f(\frac{2}{5}) = - \sqrt {\frac{2}{5}} ‧ 3 \sqrt {\frac{18}{5}} \)

\(f(6) = - \sqrt 6 ‧ 3 \sqrt 2 \)

( c , d ) = \(( \frac{2}{5} , 6 \))。