Consider the function y = 2x^3-3x^2+9x+5.

1. Find where the function is increasing, where the function is decreasing, and calculate all relative extrema (local max and local min)
2. Find where the function is concave up, where the function is concave down, and find the inflection points.

y = 2x^3-3x^2+9x+5

y' = 6x^2-6x+9
y" = 12x-6

Now, just apply the basic rules
extrema at y'=0 and y"≠0
y increasing where y' > 0
y concave up where y" > 0
inflection points at y"=0

Note: There may not be any extrema, and so on. Sometimes the required conditions do not exist.

To find where the function is increasing, decreasing, and locate the relative extrema, we need to take the derivative of the function and solve it for critical points. Let's start with the first question.

1. Finding where the function is increasing and decreasing:

To find where the function is increasing or decreasing, we need to find the critical points by setting the derivative equal to zero:

dy/dx = 6x^2 - 6x + 9 = 0

Let's solve this quadratic equation:

Set the equation equal to zero:

6x^2 - 6x + 9 = 0

Divide all terms by 3:

2x^2 - 2x + 3 = 0

Now, we can use the quadratic formula to solve for x:

x = (-b ± √(b^2 - 4ac)) / 2a

For this equation, a = 2, b = -2, and c = 3:

x = (-(-2) ± √((-2)^2 - 4 * 2 * 3)) / (2 * 2)

Simplifying this expression gives us:

x = (2 ± √(4 - 24)) / 4
x = (2 ± √(-20)) / 4

Since the discriminant is negative, there are no real solutions for x. Therefore, there are no critical points.

Since there are no critical points, the function does not have any local maxima or minima. So, the function is increasing or decreasing for all values of x.

2. Finding where the function is concave up, concave down, and locating inflection points:

To determine where the function is concave up or concave down and find the inflection points, we need to calculate the second derivative.

Taking the derivative of the given function:

d²y/dx² = d/dx(6x^2 - 6x + 9)
= 12x - 6

Now, we'll set the second derivative equal to zero to find possible inflection points:

12x - 6 = 0

Adding 6 to both sides:

12x = 6

Dividing both sides by 12:

x = 1/2

Thus, x = 1/2 is the only critical point we found, which is a possible inflection point.

Now, we need to determine the nature of the function on each side of the potential inflection point. We can test a value greater than 1/2 and a value less than 1/2 to determine if the function is concave up or concave down. Let's try x = 0 and x = 1:

For x = 0:
d²y/dx² = 12(0) - 6 = -6
Since the second derivative is negative, the function is concave down.

For x = 1:
d²y/dx² = 12(1) - 6 = 6
Since the second derivative is positive, the function is concave up.

Therefore, the function y = 2x^3 - 3x^2 + 9x + 5 is concave down for x < 1/2 and concave up for x > 1/2. The point (1/2, f(1/2)) is a potential inflection point. To determine if it is an actual inflection point, we should check the behavior of the function at x = 1/2.