Problem+39+Yutong+and+Meredith+GM

Problem #39: Meredith G-M

A(x)=x√(x+3)


 * a**) Find the intervals of increase and decrease

A'(x) = x[½(x+3)^-½] + √(x+3) A'(x) = [3(x+2)]/[2√(x+3)]

Find critical points: [3(x+2)]/[2√(x+3)] = 0 x = -2


 * **Interval** || **f'(x)** || **f(x)** ||
 * -∞, -3 || Negative || Decreasing ||
 * -3,∞ || Positive || Increasing ||

So, the graph of f(x) is decreasing over (-∞, -3) and increasing over ( -3,∞)

Ms. Sweeney: only one comment, and that is that the domain is [-3, ∞)


 * b**) Find the local minimum and local maximum

Since the first derivative, f'(x), is changing from negative to positive at x=-2, the graph f(x) has a local minimum at that point. There is no local maximum.


 * c**) Find the intervals of concavity and points of inflection

A'(x) = [3(x+2)]/[2√(x+3)] A''(x) = ¾[(x+4)/√(x+3)³]

A''(x) is always positive and never equals 0, so there is no change in concavity or inflection point. The graph f(x) is always concave upward.

d) Graph this equation