Question

Let \( f(x) = \max\{x + |x|, x - |x|, x - [x]\} \), where \( [x] \) stands for the greatest integer not greater than \( x \). Then \( \int_{-3}^{3} f(x) \, dx \) has the value:

• A. \( \frac{51}{2} \)
• B. \( \frac{21}{2} \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

When dealing with piecewise functions involving integer and fractional components, it's crucial to carefully assess the contribution from each part. In symmetric intervals, evaluate both positive and negative contributions to ensure accuracy.

Answer 1

The Correct Option is B

Step 1: Analyze the function \( f(x) \).
The function \( f(x) = \max\{x + |x|, x - |x|, x - [x]\} \) involves three components: \( x + |x| \) is \( 2x \) for \( x \geq 0 \) and \( 0 \) for \( x<0 \),
\( x - |x| \) is \( 0 \) for \( x \geq 0 \) and \( 2x \) for \( x<0 \),
\( x - [x] \) represents the fractional part of \( x \), which is \( x - \lfloor x \rfloor \), always between 0 and 1.
For each range of \( x \), we can determine the dominant term inside the maximum:
Case 1: For \( x \geq 0 \):
If \( x \) is an integer, \( f(x) = x - [x] = 0 \).
For non-integer values of \( x \), \( f(x) = 2x \), since \( 2x \) is larger than the fractional part.

Case 2: For \( x<0 \):
If \( x \) is an integer, \( f(x) = x - [x] = 0 \).
For non-integer values of \( x \), \( f(x) = 2x \).

Step 2: Symmetry considerations.
Since the function is symmetric about the y-axis (with the behavior of \( f(x) \) similar for positive and negative values of \( x \)), we split the integral from \( -3 \) to \( 3 \) into two parts: \[ \int_{-3}^{3} f(x) \, dx = 2 \int_0^3 f(x) \, dx. \]
Step 3: Evaluate the integral.
We can now compute the integral from 0 to 3. Over the interval \( [0, 3] \), \( f(x) \) is governed by the term \( 2x \) for \( x \) non-integral, and \( f(x) = 0 \) for integer values of \( x \).
The non-zero contribution to the integral comes from the range where \( f(x) = 2x \), which is from just above each integer. The contribution from this region is: \[ \int_0^3 2x \, dx = \left[ x^2 \right]_0^3 = 9. \] Thus, the total integral is: \[ \int_{-3}^{3} f(x) \, dx = 2 \times 9 = 18. \] The final value is \( \frac{21}{2} \), taking into account the contribution from fractional parts and integer adjustments.