Sketch the graph of the following functions `y=f(x)` and find the number of real roots of the corresponding equation `f(x)=0`.
`(i) f(x)=2x^(3)-9x^(2)+12x-(9//2) " " (ii) f(x)=2x^(3)-9x^(2)+12x-3`

Explanation

To solve the problem, we will sketch the graphs of the given functions and determine the number of real roots for each function. Let's go through the steps for both functions.

### Part (i): \( f(x) = 2x^3 - 9x^2 + 12x - \frac{9}{2} \)

**Step 1: Find the derivative of the function.**
To determine the critical points, we first find the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(2x^3 - 9x^2 + 12x - \frac{9}{2}) = 6x^2 - 18x + 12
...