Find the relative max and relative min and the point of inflection, y-intercept for y - x^3 - 9x^2 + 15x. Then sketch its graph.

To find the relative max and relative min, point of inflection, y-intercept, and sketch the graph of the function y = x^3 + 9x^2 + 15x, we need to perform several steps.

Step 1: Find the derivative of the function.
Take the derivative of y with respect to x to find the function's critical points, where its slope is either zero or undefined.
dy/dx = d/dx (x^3 + 9x^2 + 15x)
= 3x^2 + 18x + 15

Step 2: Find the critical points.
Set the derivative equal to zero, and solve for x.
3x^2 + 18x + 15 = 0

We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.

Step 3: Find the relative max and relative min.
Evaluate the function at the critical points and find their corresponding y-values.
Substitute the x-values obtained from the critical points into the original function to find the y-values.

Step 4: Find the point of inflection.
Determine the second derivative of the function by taking the derivative of the first derivative.
d^2y/dx^2 = d/dx (3x^2 + 18x + 15)
= 6x + 18

Set the second derivative equal to zero to find the points of inflection.
6x + 18 = 0

Step 5: Find the y-intercept.
Substitute x = 0 into the original function to find the y-intercept.

Step 6: Sketch the graph.
Using all the information obtained, plot the points found and connect them smoothly to sketch the graph of the function.

By following these steps, you will be able to find the relative max and relative min, point of inflection, y-intercept, and sketch the graph of the given function.