How do you find the local max and min for #f(x) = 7x + 9x^(-1)#?

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Answer 1 · 2016-11-29T20:55:22

To find the local extrema we find the points where the first derivative is null and study the sign of the second derivative

#f(x) =7x+9x^(-1)#

#f'(x) =7 -9x^(-2)#

#f''(x) =18x^(-3)#

Find the values of #x# where #f'(x)=0#

#7-9/x^2 = 0#

#7x^2-9 = 0#

#x=+-3/sqrt(7)#

In both points #f''(x) !=0# so these are local extrema, namely:

1) around #x=-3/sqrt(7)# the second derivative is negative and then the point is a local maximum.

2) around #x=+3/sqrt(7)# the second derivative is positive and then the point is a local minimum.

graph{7x+9/x [-47.3, 47.2, -73.6, 73.6]}