g(x) = -x^5 + 11x^4 - 43x^3 + 69x^2 - 36x

d) Clearly mark these points on a graph
- At the first turning point
- At the only root with order two

http://www.wolframalpha.com/input/?i=y%3D+-x%5E5+%2B+11x%5E4+-+43x%5E3+%2B+69x%5E2+-+36x

Note in the alternate forms of the equation, the (x-3)^2 factor which indicates a double root.

To find the first turning point of the function g(x), we need to find the value of x where the derivative of g(x) is equal to zero.

Let's start by finding the derivative of g(x):

g'(x) = d/dx(-x^5 + 11x^4 - 43x^3 + 69x^2 - 36x)

Using the power rule and the constant multiple rule, we can find:

g'(x) = -5x^4 + 44x^3 - 129x^2 + 138x - 36

Next, we'll find the values of x where g'(x) = 0.

-5x^4 + 44x^3 - 129x^2 + 138x - 36 = 0

Unfortunately, solving this polynomial equation directly is quite complex. However, we can use graphical methods or approximation techniques to find the turning points.

To mark these points on a graph, we'll plot g(x) and identify the x-values where the curve changes its direction (turning points).

Now, let's plot the graph of g(x) and mark the turning points:

Note: Please bear with me as this is a text-based platform, so I won't be able to provide an actual graph. I will describe it step-by-step.

1. Start with a coordinate plane with x and y axes.
2. Choose a suitable range for the x-axis to include the values where the turning point and the root with order two are expected.
3. Calculate the corresponding y-values for some x-values in that range using the equation g(x) = -x^5 + 11x^4 - 43x^3 + 69x^2 - 36x.
4. Plot the points (x, y) on the graph.
5. Observe the shape of the graph. At the turning point, the curve changes from increasing to decreasing or vice versa. For the only root with order two, the curve intersects the x-axis at that point.
6. Identify the first turning point and mark it on the graph with a point or an open circle.
7. Identify the only root with order two and mark it on the graph with a filled circle or a solid dot.

The graph will give you a visual representation of the function g(x) and its turning points.

To find the first turning point, we need to find the value of x where the derivative of g(x) is equal to zero. The derivative of g(x) will give us the slope of the original function at any given point.

First, let's find the derivative of g(x) with respect to x:

g'(x) = -5x^4 + 44x^3 - 129x^2 + 138x - 36

Now, we'll set g'(x) equal to zero and solve for x:

-5x^4 + 44x^3 - 129x^2 + 138x - 36 = 0

At this point, you can either solve this equation by factoring, using the rational root theorem, or by using numerical methods. Once you find the values of x that satisfy this equation, you can substitute them back into g(x) to find the corresponding y-values.

Now let's find the only root with order two. A root with order two means that when we solve for x, we will get two identical solutions. To find it, we need to find the values of x where g(x) is equal to zero.

Setting g(x) equal to zero:

-g(x) = -(-x^5 + 11x^4 - 43x^3 + 69x^2 - 36x) = x^5 - 11x^4 + 43x^3 - 69x^2 + 36x = 0

Just like before, we can solve this equation either by factoring, using the rational root theorem, or by using numerical methods. Once you find the values of x that satisfy this equation, you can mark them on a graph with their corresponding y-values.

After finding the values for both the first turning point and the only root with order two, you can mark them on a graph.