Analyze and graph the following function: y = x(9-x^2)^(1/2)

Question

Analysis must include domain and range, symmetry, x and y intercepts, all asymptotes, first and second derivatives, critical numbers, intervals at which the function is increasing and decreasing as well as concave up and concave down, all relative extrema, and all possible points of inflection.

Answer 1

Domain: -3 to 3, because 9-x²≥0 so x²≤9, |x|≤3.

Range: -4.5 to 4.5, because d(x√(9-x²))/dx=-x²/√(9-x²)+√(9-x²)=0 (max/min) when -x²+9-x²=0=9-2x².

x=±√(9/2)=±3/√2=±3√2/2, and y=±(3√2/2)√(9-9/2)=±9/2=±4.5.

Intercepts: y=0 when x=±3 so x intercepts are at -3 and 3. When x=0 y=0, so y intercept is at 0.

No asymptotes but graph only exists between and at the x intercepts.

Minimum (concave up) at (-3√2/2, -4.5) and maximum (concave down) at (3√2/2, 4.5).

 

Graph to follow (internal system error prevents upload—ads interfere with upload)...

Description of graph: starts at (-3,0), dips to minimum at (-3√2/2,-4.5), rises to pass through (0,0) then continues to rise to maximum at (3√2/2,4.5) and then falls to its termination at (3,0). The left and right halves of the graph are symmetrical but inverted—y is an odd function: y(x)=-y(-x).