F(x) = x^5 In x What is Che domain of the function? Find the interval(s) on whic
ID: 2859569 • Letter: F
Question
F(x) = x^5 In x What is Che domain of the function? Find the interval(s) on which f is increasing. (Enter your answer using interval notation.) Find the interval(s) on which f is decreasing. (Enter your answer using interval notation.) Find the x-coordinate(s) of any local minima. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE. Local minima at x = Find the x-coordinate(s) of any local maxima. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Local maxima atx = Find the interval(s) on which f's concave up. (Enter your answer using interval notation.) Find the interval(s) on which f's concave down. (Enter your answer using interval notation.) Find the x-coordinate(s) of any inflection points. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Inflection point(s) at x =Explanation / Answer
f(x)=x5(ln x)
a)domain=(0,)
b)differentite with respect to x
f '(x)=5x4(ln x) +x5(1/x)
f '(x)=5x4(ln x) +x4
f '(x)=x4(5(ln x) +1)
function is increasing when f '(x)>0
=>(5(ln x) +1)>0
=>lnx>-1/5
=>x>e-1/5
function is increasing when x =(e-1/5,infinity)
function is decreasing when f '(x)<0
=>(5(ln x) +1)<0
=>lnx<-1/5
=>x<e-1/5
function is decreasing when x =(0,e-1/5)
c) x =e-5
f '(x) changes its sign from negative to positive
=>local minima occurs at x =e-5
f(e-5)=(e-5)5(ln e-5)
f(e-5)=-5e-25
coordiate of local minima are (e-5,-5e-25)
d)f '(x)=x4(5(ln x) +1)
differentiate with respect to x
f '(x)=4x3(5(ln x) +1) +x4(5(1/x) +0)
f '(x)=4x3(5(ln x) +1) +(5x3)
f '(x)=x3(20(ln x) +4 +5)
f '(x)=x3(20(ln x) +9)
for inflection point f "(x)=0
=>x3(20(ln x) +9)=0
=>(20(ln x) +9)=0
=>lnx =-9/20
=>x =e-9/20
concave up when f "(x)>0
=>(e-9/20,infinity)
concave down when f "(x)<0
=>(0,e-9/20)
inflection point
x=e-9/20
f(e-9/20)=(e-9/20)5(ln e-9/20)
f(e-9/20)=(-9/20)e-9/4
inflection point =(e-9/20,(-9/20)e-9/4)
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