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Kerjakan dengan cara. Itu xkuadrat dikurangi dengan x

The problem asks to determine for which range of \( x \) the curve \( y = 2^{x^2 - x} \) is located below the line \( y = 64 \).

### Step-by-Step Solution:

1. **Equation of the Curve:**
   \[
   y = 2^{x^2 - x}
   \]

2. **Equation of the Line:**
   \[
   y = 64
   \]

   Since \( 64 \) can be written as \( 2^6 \), we set the equation of the curve equal to \( 64 \) to find where the curve intersects the line:

   \[
   2^{x^2 - x} = 2^6
   \]

3. **Solving the Equation:**
   Since the bases are the same, we equate the exponents:

   \[
   x^2 - x = 6
   \]

   This leads to the quadratic equation:

   \[
   x^2 - x - 6 = 0
   \]

4. **Factoring the Quadratic Equation:**
   \[
   (x - 3)(x + 2) = 0
   \]

   So, the solutions are:

   \[
   x = 3 \quad \text{or} \quad x = -2
   \]

5. **Interval Analysis:**
   Now, we analyze the intervals determined by \( x = -2 \) and \( x = 3 \) to see where \( y = 2^{x^2 - x} < 64 \).

   - For \( x < -2 \), let's pick \( x = -3 \):
     \[
     x^2 - x = (-3)^2 - (-3) = 9 + 3 = 12 \quad \text{and} \quad 2^{12} > 64
     \]
     So, the curve is above the line.

   - For \( -2 < x < 3 \), let's pick \( x = 0 \):
     \[
     x^2 - x = 0^2 - 0 = 0 \quad \text{and} \quad 2^0 = 1 < 64
     \]
     So, the curve is below the line.

   - For \( x > 3 \), let's pick \( x = 4 \):
     \[
     x^2 - x = 4^2 - 4 = 16 - 4 = 12 \quad \text{and} \quad 2^{12} > 64
     \]
     So, the curve is above the line.

   The curve \( y = 2^{x^2 - x} \) is below the line \( y = 64 \) for \( -2 < x < 3 \).

### Answer:
The correct choice is **c. \( -2 < x < 3 \)**.

Would you like more details or explanations? 

Here are 5 related questions:
1. How would the answer change if the line equation was \( y = 32 \) instead of \( y = 64 \)?
2. Can you derive the discriminant of the quadratic equation \( x^2 - x - 6 = 0 \) to verify the nature of the roots?
3. How does the behavior of the curve change if the exponent in the curve equation was \( x^2 + x \) instead of \( x^2 - x \)?
4. What is the significance of finding the intervals in which the curve lies below or above a particular line?
5. How would you graphically represent the solution to this problem?

**Tip:** Always ensure that the intervals derived from solving inequalities are correctly tested with sample points to confirm where the inequality holds true.

Learn how to determine for which range of x the curve y = 2^(x^2 - x) is located below the line y = 64 with a detailed step-by-step solution. This problem covers exponential functions, inequalities, and quadratic equations. Suitable for high school students.

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