https://file.aiquickdraw.com/mathbot/file/1729256577541dfm6qcat.jpg

The problem in the image is:

"Grafik fungsi \( y = ax^3 - 2x^2 + 2ax + m - 3 \) menyinggung sumbu \( x \). Tentukan nilai \( m \) yang memenuhi."

Translation:
"The graph of the function \( y = ax^3 - 2x^2 + 2ax + m - 3 \) touches the \( x \)-axis. Determine the value of \( m \) that satisfies this condition."

### Solution:
If the graph of the function touches the \( x \)-axis, it means the function has a double root (i.e., the discriminant must be zero at that point). We need to analyze when the cubic function has a root with multiplicity 2.

1. **Function:** \( y = ax^3 - 2x^2 + 2ax + m - 3 \).
2. **Condition:** The graph touches the \( x \)-axis, so \( y = 0 \) has a repeated root.

Let's proceed with the necessary steps:

1. **Find the derivative of \( y \)**:
   \[
   y' = \frac{d}{dx}(ax^3 - 2x^2 + 2ax + m - 3) = 3ax^2 - 4x + 2a
   \]
   
2. **Condition for touching the \( x \)-axis:

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

Determine the value of m in the function y = ax^3 - 2x^2 + 2ax + m - 3 where the graph touches the x-axis. This problem covers key calculus concepts, such as the derivative of cubic functions and conditions for tangency.

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