Question

F(x) = x^5 ln x
A) Find the interval on which f is increasing Find the interval on which f is decreasing
B) Find the local minimum value of f
C) find the inflection point (x,y)
Find the interval in which f is concave up
Find the interval in which f is concave down

Answer 1

f' = 5x^4 lnx + x^4 = x^4(5lnx + 1)
f'' = 20x^3 lnx + 5x^3 + 4x^3 = x^3(20lnx + 9)

f' = 0 at x = 0 or e^(-1/5)
f'' = 0 at x=0 or e^(-9/20)

as you know
f increasing where f' > 0
min where f'=0 and f'' not zero
concave up where f'' > 0