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The problem asks to determine the values of \( m \) and \( n \) such that the graph of the function

\[
y = nx^3 - 2x^2 + 2nx + m - 3
\]

is tangent to the x-axis. To solve this, let's break it down into steps:

### 1. Condition for Tangency
For a graph to be tangent to the x-axis at a point, the following must hold:
- The y-value at that point must be zero (so \( y = 0 \)).
- The slope (or derivative) at that point must also be zero.

Let’s denote the tangency point as \( x = a \). We need:
- \( f(a) = 0 \) (the function equals zero at this point)
- \( f'(a) = 0 \) (the derivative equals zero at this point)

### 2. Expression for the Derivative
We first find the derivative \( y' \) of the function:
\[
y = nx^3 - 2x^2 + 2nx + m - 3
\]

Differentiating with respect to \( x \):
\[
y' = 3nx^2 - 4x + 2n
\]

### 3. Apply the Tangency Condition
At the point of tangency, \( x = a \), both the function and its derivative are zero:
- \( f(a) = 0 \)
- \( f'(a) = 0 \)

By solving these two equations, we can find the values of \( m \) and \( n \) that satisfy these conditions.

Would you like to proceed with solving these equations? Or do you have any specific questions about this step?

### Related Questions:
1. What is the significance of a graph being tangent to the x-axis?
2. How do we determine the point of tangency mathematically?
3. What is the relationship between the derivative of a function and the slope of its graph?
4. Why does the derivative need to be zero at the tangency point?
5. How does the value of \( m \) shift the graph of the cubic function vertically?

**Tip:** When finding where a graph is tangent to an axis, always check both the function's value and its derivative at the point of tangency.

Grafik fungsi y = nx^3 - 2x^2 + 2nx + m - 3 menyinggung sumbu x. Tentukan nilai m dan n yang memenuhi.

Determine the values of m and n such that the graph of the cubic function y = nx^3 - 2x^2 + 2nx + m - 3 is tangent to the x-axis. Learn about the conditions for tangency and the derivative test. Suitable for students in Grades 10-12.

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