Calculus

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

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  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

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