To solve this problem, we are asked to find the points on the graph of the function $f(x) = 2x^2 - 3x + 3$ where the tangent line is horizontal. A tangent line is horizontal when its slope is zero. The slope of the tangent line is given by the derivative of the function $f(x)$.

Steps:

1. Find the derivative of $f(x)$:

   The derivative of the function $f(x) = 2x^2 - 3x + 3$ is: $f'(x) = \frac{d}{dx}(2x^2 - 3x + 3)$ Using basic differentiation rules: $f'(x) = 4x - 3$

2. Set the derivative equal to 0:

   To find where the tangent line is horizontal, we set the derivative equal to zero and solve for $x$: $4x - 3 = 0$ Solving for $x$: $4x = 3$ $x = \frac{3}{4}$

3. Find the corresponding $y$-coordinate:

   To find the point where the tangent is horizontal, substitute $x = \frac{3}{4}$ into the original function: $f\left(\frac{3}{4}\right) = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 3$ Simplifying: $f\left(\frac{3}{4}\right) = 2\left(\frac{9}{16}\right) - \frac{9}{4} + 3$ $f\left(\frac{3}{4}\right) = \frac{18}{16} - \frac{9}{4} + 3 = \frac{9}{8} - \frac{18}{8} + \frac{24}{8}$ $f\left(\frac{3}{4}\right) = \frac{15}{8}$

Thus, the point where the tangent line is horizontal is $\left( \frac{3}{4}, \frac{15}{8} \right)$.

Final Answer:

The point where the tangent line is horizontal is $\left( \frac{3}{4}, \frac{15}{8} \right)$.

Would you like more details on any part of the solution or have any questions?

Here are 5 related questions to consider:

1. How do you find the derivative of higher-order polynomials?
2. What is the geometric interpretation of a horizontal tangent line?
3. How does the shape of the graph change near points where the tangent is horizontal?
4. What if a function has multiple points where the tangent is horizontal?
5. How do you determine if a critical point is a maximum, minimum, or neither?

Tip: The derivative represents the slope of the tangent line at any point on a function, and setting it equal to zero can help find important points like maxima, minima, or points of inflection.