What is the extreme f(x)=-5x^{+30x+1} ?

The extreme f(x)=-5x^{+30x+1} is

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extreme f(x)=-5x^{+30x+1}

extreme points

f (x) = - 5 x^{+ 30 x + 1}

Maximum (0, 0), Minimum - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )}, - 5 (- \frac{1}{30 W _{- 1} ( - \frac{e}{30} )})^{- \frac{1}{W _{- 1} ( - \frac{e}{30} )} + 1}, Maximum - \frac{1}{30 W _{0} ( - \frac{e}{30} )}, - 5 (- \frac{1}{30 W _{0} ( - \frac{e}{30} )})^{- \frac{1}{W _{0} ( - \frac{e}{30} )} + 1}

Solution steps

Find the critical points:

x = 0, x = - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )}, x = - \frac{1}{30 W _{0} ( - \frac{e}{30} )}

Domain of

- 5 x^{+ 30 x + 1} : x \geq 0

Find intervals:

Decreasing

: 0 < x < - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )},

Increasing

: - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )} < x < - \frac{1}{30 W _{0} ( - \frac{e}{30} )},

Decreasing

: - \frac{1}{30 W _{0} ( - \frac{e}{30} )} < x < \infty

Plug the extreme point

x = 0

into

- 5 x^{+ 30 x + 1} \Rightarrow y = 0

Maximum (0, 0)

Plug the extreme point

x = - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )}

into

- 5 x^{+ 30 x + 1} \Rightarrow y = - 5 (- \frac{1}{30 W _{- 1} ( - \frac{e}{30} )})^{- \frac{1}{W _{- 1} ( - \frac{e}{30} )} + 1}

Minimum - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )}, - 5 (- \frac{1}{30 W _{- 1} ( - \frac{e}{30} )})^{- \frac{1}{W _{- 1} ( - \frac{e}{30} )} + 1}

Plug the extreme point

x = - \frac{1}{30 W _{0} ( - \frac{e}{30} )}

into

- 5 x^{+ 30 x + 1} \Rightarrow y = - 5 (- \frac{1}{30 W _{0} ( - \frac{e}{30} )})^{- \frac{1}{W _{0} ( - \frac{e}{30} )} + 1}

Maximum - \frac{1}{30 W _{0} ( - \frac{e}{30} )}, - 5 (- \frac{1}{30 W _{0} ( - \frac{e}{30} )})^{- \frac{1}{W _{0} ( - \frac{e}{30} )} + 1}

Maximum (0, 0), Minimum - \frac{1}{30 W _{- 1} ( - \frac{e}{30} )}, - 5 (- \frac{1}{30 W _{- 1} ( - \frac{e}{30} )})^{- \frac{1}{W _{- 1} ( - \frac{e}{30} )} + 1}, Maximum - \frac{1}{30 W _{0} ( - \frac{e}{30} )}, - 5 (- \frac{1}{30 W _{0} ( - \frac{e}{30} )})^{- \frac{1}{W _{0} ( - \frac{e}{30} )} + 1}

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