Functions?

Question

Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
#h(x) = 7sqrt(x) e^(−x)#
Increasing: ???
decreasing:???

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Answer 1 · 2017-07-16T06:35:55

The function #h(x)=7sqrtxe^(-x)# is increasing in open interval #(1/2,oo)# and decreasing in open interval #(0,1/2)#

We have #h(x)=7sqrtxe^(-x)#. Observe that it is not defined for #x<0#.

Now as #(dh)/(dx)=7[e^(-x)/(2sqrtx)-sqrtxe^(-x)]=(7e^(-x))/(2sqrtx)(1-2x)#

#(dh)/(dx) >0# for #x<1/2# and #(dh)/(dx) <0# for #x>1/2#

and hence the function #h(x)=7sqrtxe^(-x)#

is increasing in open interval #(0,1/2)#

and decreasing in open interval #(1/2,oo)#

graph{7sqrtxe^(-x) [-3.063, 6.937, -1.06, 3.94]}