Recall that the inflection points of a graph can be found by setting the second derivative of the function equal to 0 and solving for x. Also, the intervals of concavity can be found by determining where the second derivative is positive and negative.
First, let's find the second derivative of the function by using the power rule for derivatives twice,
f(x) = 3x + (x + 2)^(3/5)
f '(x) = 3 + (3/5)(x + 2)^(-2/5)
f ''(x) = (-6/25)(x + 2)^(-7/5)
Now set this equation equal to 0 and solve for x to find the inflection points,
(-6/25)(x + 2)^(-7/5) = 0
(x + 2)^(-7/5) = 0
x + 2 = 0
x = -2
So (-2,0) is an inflection point of the graph.
Now we can plug in values greater than and less than -2 to find where the graph his positive and negative,
Let's use -3 and -1
f ''(x) = (-6/25)(x + 2)^(-7/5)
f ''(-3) = (-6/25)(-3 + 2)^(-7/5) = (-6/25)(-1)^(-7/5) = (-6/25)(-1) = 6/25
Since 6/25 is positive, the graph is concave up on the interval (-∞,-2)
f ''(x) = (-6/25)(x + 2)^(-7/5)
f ''(-1) = (-6/25)(-1 + 2)^(-7/5) = (-6/25)(1)^(-7/5) = (-6/25)(1) = -6/25
Since -6/25 is negative, the graph is concave down on the interval (-2,∞)
