Find the critical points of the function and use the First Derivative Test to de

Question

Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Enter your answers as a comma-separated as a comma-separated list. If an answer does not exist, enter DNE.) f(x) = 5x + 2x^-1 + 7, (x > 0) local minimum c = ___________ local maximum c = ___________ Determine the intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. Enter EMPTY or o slash for the empty set.) increasing ___________ decreasing ___________

Explanation / Answer

given f(x)=5x+2x-1+7, x>0

domain (0,)

differentiate wiith respect to x

f '(x)=(5*1)+2(-1)x-1-1+0

f '(x)=5-2x-2

f '(c)=5-2c-2

for critical point f '(c) =0

5-2c-2=0

2c-2=5

c2=2/5

c=(2/5)

for c <(2/5) , f '(c) <0

for c >(2/5) , f '(c) >0

f '(c) changes from negative to positive at c =(2/5)

so function has local minimum at (2/5)

local maximum DNE

increasing when f '(c)>0

=>increasing in ((2/5) ,)

decreasing when f '(c)<0

=>decreasing in (0,(2/5))

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