Question

For what values of x is `f(x)=-2x- (x+3)^ (-2/3)` concave or convex?

Answer 1

`x in (-infty,-3)`: `f(x)` is concave down

`x in (-3,+infty)`: `f(x)` is concave down

Explanation:

Find the first derivative, which is the slope of the curve

`f'(x)=-2+2/3(x+3)^(-5/3)`

The second derivative, is the rate of change of the curve

`f''(x)=-10/9(x+3)^(-8/3)=(-10)/(9root(3)((x+3)^8))`

Whenever `f''(x) < 0`, the curve is concave downward (like a frown face) and whenever `f''(x) > 0`, the curve is concave upward (like a smiley face).

Testing points around `x=-3` (where `f''(x)` divides by zero).

`x < -3`: All points are negative; Concave down

`x > -3`: All points are negative; Concave down