What is the extreme x^{25/3}-6x^{19/3} ?

The extreme x^{25/3}-6x^{19/3} is

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extreme x^{25/3}-6x^{19/3}

extreme

Maximum - \frac{11 4 ^{\frac{1}{2}}}{5}, (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}, Saddle (0, 0), Minimum \frac{11 4 ^{\frac{1}{2}}}{5}, (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}

Solution steps

Find the critical points:

x = - \frac{11 4 ^{\frac{1}{2}}}{5}, x = 0, x = \frac{11 4 ^{\frac{1}{2}}}{5}

Domain of

x^{\frac{25}{3}} - 6 x^{\frac{19}{3}} : - \infty < x < \infty

Find intervals:

Increasing

: - \infty < x < - \frac{11 4 ^{\frac{1}{2}}}{5},

Decreasing

: - \frac{11 4 ^{\frac{1}{2}}}{5} < x < 0,

Decreasing

: 0 < x < \frac{11 4 ^{\frac{1}{2}}}{5},

Increasing

: \frac{11 4 ^{\frac{1}{2}}}{5} < x < \infty

Plug the extreme point

x = - \frac{11 4 ^{\frac{1}{2}}}{5}

into

x^{\frac{25}{3}} - 6 x^{\frac{19}{3}} \Rightarrow y = (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}

Maximum - \frac{11 4 ^{\frac{1}{2}}}{5}, (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}

Plug the extreme point

x = 0

into

x^{\frac{25}{3}} - 6 x^{\frac{19}{3}} \Rightarrow y = 0

Saddle (0, 0)

Plug the extreme point

x = \frac{11 4 ^{\frac{1}{2}}}{5}

into

x^{\frac{25}{3}} - 6 x^{\frac{19}{3}} \Rightarrow y = (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}

Minimum \frac{11 4 ^{\frac{1}{2}}}{5}, (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}

Maximum - \frac{11 4 ^{\frac{1}{2}}}{5}, (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (- \frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}, Saddle (0, 0), Minimum \frac{11 4 ^{\frac{1}{2}}}{5}, (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{25}{3}} - 6 (\frac{11 4 ^{\frac{1}{2}}}{5})^{\frac{19}{3}}