The document finds the critical numbers and points of inflection for the curve y=x^(1/3)*(x+3)^(2/3). It takes the derivative of the function and sets it equal to 0. This results in the critical point and sole point of inflection being at x = -1.
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Find the critical numbers and points of inflection for the curve y=x.pdf
1. Find the critical numbers and points of inflection for the curve y=x^(1/3)*(x+3)^(2/3).
Solution
The function we have is y=x^(1/3)*(x+3)^(2/3).
y' = [x^(1/3)]'*(x+3)^(2/3) + x^(1/3)*[(x+3)^(2/3)]'
y' = (1/3)*x^(-2/3)*(x+3)^(2/3) + x^(1/3)*(2/3)*(x+3)^(-1/3)
y' = (1/3)*[(x+3)/x]^(2/3) + (2/3)*[x/(x + 3)]^(1/3)
y' = [(1/3)*(x+3) + (2/3)*x]/x^(2/3)*(x + 3)^(1/3)
y' = (x + 1)/x^(2/3)*(x + 3)^(1/3)
The critical point lies where y' = 0, here this occurs at x = -1
The critical point and the point of inflection is at x = -1.