To find the value of $x$ in the interval $1 < x < 2$ where $f(x)$ attains its absolute maximum, we proceed by analyzing the behavior of the derivative $f'(x) = 2\sin(\pi x)$.

Step 1: Identify critical points

A critical point occurs where $f'(x) = 0$. To find these points, set the derivative equal to zero:

$f'(x) = 2\sin(\pi x) = 0$

This simplifies to:

$\sin(\pi x) = 0$

The sine function equals zero at integer multiples of $\pi$, i.e., when $\pi x = n\pi$, where $n$ is an integer. Thus:

$x = n$

In the interval $1 < x < 2$, the only integer is $x = 1$, but since this is not strictly inside the interval, we exclude it. However, we do need to check for points where the derivative changes sign within this interval.

Step 2: Analyze the behavior of $f'(x)$ in $1 < x < 2$

We will check the sign of $f'(x)$ in the interval $1 < x < 2$ by evaluating $\sin(\pi x)$ at some representative points.

• At $x = 1.5$: $\sin(\pi \times 1.5) = \sin\left(\frac{3\pi}{2}\right) = -1$ So $f'(1.5) = 2 \times (-1) = -2$, which means the function is decreasing at $x = 1.5$.

Since $\sin(\pi x)$ is negative in the interval $1 < x < 2$ (excluding $x = 1$), the derivative is negative throughout this interval, indicating that $f(x)$ is decreasing.

Step 3: Evaluate the endpoints

The function could attain its absolute maximum at the boundary points. To determine this, we check the values of $f(x)$ at the endpoints of the interval $1 < x < 2$. However, since the function is decreasing in the entire interval, the absolute maximum will occur at the left endpoint of the interval, which is just after $x = 1$.

Therefore, the absolute maximum of $f(x)$ in the interval $1 < x < 2$ occurs as $x$ approaches 1 from the right.

Final Answer:

The absolute maximum occurs as $x \to 1^+$ in the interval $1 < x < 2$.

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Would you like further clarification on any of the steps?

Here are 5 related questions to expand your understanding:

1. How would the result change if the interval were $0 < x < 2$?
2. What is the general behavior of the function $f(x)$ given its derivative?
3. How would you determine if the maximum is absolute or local based on $f'(x)$?
4. What role do critical points outside the interval play in determining maxima or minima?
5. How does the periodic nature of sine affect the behavior of the derivative?

Tip: To analyze critical points, always check the derivative's sign changes, as they indicate where a function increases or decreases.