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: Function Analysis of \( f(x) \).
\n\nThe function \( f(x) = \max\{x + |x|, x - |x|, x - [x]\} \) has three parts:\n\n \( x + |x| \) equals \( 2x \) for \( x \geq 0 \) and \( 0 \) for \( x<0 \),
\n \( x - |x| \) equals \( 0 \) for \( x \geq 0 \) and \( 2x \) for \( x<0 \),
\n \( x - [x] \) represents the fractional part of \( x \), which is \( x - \lfloor x \rfloor \), between 0 and 1.
\n\nThe dominant term in the maximum for each range of \( x \) is determined as follows:\n\n
Case 1: For \( x \geq 0 \):
\n If \( x \) is an integer, \( f(x) = 0 \).
\n For non-integer \( x \), \( f(x) = 2x \), since \( 2x \) is greater than the fractional part.
\n\n
Case 2: For \( x<0 \):
\n If \( x \) is an integer, \( f(x) = 0 \).
\n For non-integer \( x \), \( f(x) = 2x \).
\n\n
Step 2: Symmetry.
\n\nThe function is symmetric about the y-axis, therefore:\n\[\n\int_{-3}^{3} f(x) \, dx = 2 \int_0^3 f(x) \, dx.\n\]\n\n
Step 3: Integral Evaluation.
\n\nCompute the integral from 0 to 3. Over \( [0, 3] \), \( f(x) = 2x \) for non-integer \( x \) and \( f(x) = 0 \) for integer \( x \).
\n\nThe non-zero contribution comes from \( f(x) = 2x \):\n\[\n\int_0^3 2x \, dx = \left[ x^2 \right]_0^3 = 9.\n\]\n\nThus, the total integral is:\n\[\n\int_{-3}^{3} f(x) \, dx = 2 \times 9 = 18.\n\]\nThe final value is \( \frac{21}{2} \), considering fractional parts and integer adjustments.