We take the derivative f ′ ( x ) = 9 7 ( x 2 7 + x − 5 7 ) = 9 7 ⋅ x + 1 x 5 7 {\displaystyle f'(x)={\frac {9}{7}}(x^{\frac {2}{7}}+x^{-{\frac {5}{7}}})={\frac {9}{7}}\cdot {\frac {x+1}{x^{\frac {5}{7}}}}} . The function therefore has two critical points: at x = − 1 , 0 {\displaystyle x=-1,0} . Now using a chart of signs shows that f ′ ( x ) {\displaystyle f'(x)} exactly when x ∈ ( − 1 , 0 ) {\displaystyle x\in (-1,0)} . We conclude that f ( x ) is decreasing on ( − 1 , 0 ) and increasing on ( − ∞ , − 1 ) ∪ ( 0 , + ∞ ) {\displaystyle \color {blue}f(x){\text{ is decreasing on }}(-1,0){\text{ and increasing on }}(-\infty ,-1)\cup (0,+\infty )} .
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The function therefore has two critical points: at 
. Now using a chart of signs shows that 
exactly when 
. We conclude that 
.
