answer:1. Define the function ( f(x) = lg left( sqrt{1 + pi^2 x^2} - pi x right) + pi ). 2. Given that ( f(m) = 3 ), we need to find ( f(-m) ). 3. Let's consider the function inside the logarithm: [ y(x) = lg left( sqrt{1 + pi^2 x^2} - pi x right) ] 4. Notice that ( y(x) ) is an odd function. To demonstrate this, consider: [ g(x) = sqrt{1 + pi^2 x^2} ] [ y(x) = lg left( g(x) - pi x right) ] For ( y(-x) ): [ y(-x) = lg left( sqrt{1 + pi^2 (-x)^2} - pi (-x) right) ] [ y(-x) = lg left( sqrt{1 + pi^2 x^2} + pi x right) ] 5. Since (sqrt{1 + pi^2 x^2}) is the same for (x) and (-x), we have: [ y(-m) = lg left( sqrt{1 + pi^2 m^2} + pi m right) ] And, [ y(m) = lg left( sqrt{1 + pi^2 m^2} - pi m right) ] 6. Simplify using the property of logarithms: [ y(m) + y(-m) = lg left( sqrt{1 + pi^2 m^2} - pi m right) + lg left( sqrt{1 + pi^2 m^2} + pi m right) ] 7. Use the logarithm addition property, (lg(a) + lg(b) = lg(ab)): [ y(m) + y(-m) = lg left( left(sqrt{1 + pi^2 m^2} - pi m right) left( sqrt{1 + pi^2 m^2} + pi m right) right) ] Simplify further: [ left( sqrt{1 + pi^2 m^2} - pi m right) left( sqrt{1 + pi^2 m^2} + pi m right) = (1 + pi^2 m^2) - (pi m)^2 ] [ = 1 + pi^2 m^2 - pi^2 m^2 = 1 ] Therefore: [ y(m) + y(-m) = lg(1) = 0 ] 8. Since: [ f(x) = y(x) + pi ] Then: [ f(m) = y(m) + pi ] [ f(-m) = y(-m) + pi ] 9. Adding (f(m)) and (f(-m)): [ f(m) + f(-m) = (y(m) + pi) + (y(-m) + pi) ] [ f(m) + f(-m) = y(m) + y(-m) + 2pi = 0 + 2pi = 2pi ] 10. Given ( f(m) = 3 ): [ 3 + f(-m) = 2pi ] 11. Solve for ( f(-m) ): [ f(-m) = 2pi - 3 ] # Conclusion: [ boxed{2pi-3} ]

answer:**Step 1: Find the largest number in each column.** - First column: max(12, 14, 10, 16, 9, 11) = 16 - Second column: max(9, 8, 4, 5, 3, 2) = 9 - Third column: max(5, 19, 6, 21, 7, 8) = 21 - Fourth column: max(4, 15, 8, 18, 13, 10) = 18 - Fifth column: max(7, 11, 12, 2, 5, 4) = 12 - Sixth column: max(3, 10, 14, 1, 6, 9) = 14 **Step 2: Check if these numbers are the smallest in their respective rows.** - For 16 in the first column, row 4: Row 4 is (16, 5, 21, 18, 2, 1). The smallest number is 1, not 16. - For 9 in the second column, row 1: Row 1 is (12, 9, 5, 4, 7, 3). The smallest number is 3, not 9. - For 21 in the third column, row 4: Row 4 is (16, 5, 21, 18, 2, 1). The smallest number is 1, not 21. - For 18 in the fourth column, row 4: Row 4 is (16, 5, 21, 18, 2, 1). The smallest number is 1, not 18. - For 12 in the fifth column, row 3: Row 3 is (10, 4, 6, 8, 12, 14). The smallest number is 4, not 12. - For 14 in the sixth column, row 3: Row 3 is (10, 4, 6, 8, 12, 14). The smallest number is 4, not 14. **Conclusion:** There is no number in the array that meets the criteria of being both the largest in its column and the smallest in its row. Thus, the correct answer is text{None}. The final answer is boxed{E}

answer:1. Examine the expression inside the absolute value: ( g(x) = x^2 - 4 ). 2. On the interval ( [-2, 2] ), compute ( g(x) ) at critical points: - ( g(-2) = (-2)^2 - 4 = 4 - 4 = 0 ) - ( g(0) = 0^2 - 4 = -4 ) - ( g(2) = 2^2 - 4 = 4 - 4 = 0 ) 3. Analyze ( g(x) ) between these points: - The function ( g(x) ) is a parabola opening upwards (since the coefficient of ( x^2 ) is positive), hitting a minimum at ( x = 0 ). - The minimum value of ( g(x) ) on the given interval is ( -4 ) and the maximum is ( 0 ). 4. Apply the absolute value: - The absolute value conversion modifies the output values: ( |g(x)| ) changes the minimum ( -4 ) to ( 4 ) and preserves the zero values. - Thus, ( |g(x)| ) has a minimum of 0 and a maximum of 4 on ( [-2, 2] ). Therefore, the range of ( f(x) ) on ( [-2, 2] ) is ( boxed{[0, 4]} ).

answer:Let's call the number x. According to the problem, the sum of the number and its square is 132. So we can write the equation as: x + x^2 = 132 Now, let's rearrange the equation to form a quadratic equation: x^2 + x - 132 = 0 To solve this quadratic equation, we can either factor it, complete the square, or use the quadratic formula. Let's try to factor it first: We are looking for two numbers that multiply to -132 and add up to 1 (the coefficient of x). These two numbers are 11 and -12 because 11 * (-12) = -132 and 11 + (-12) = -1. However, we need the numbers to add up to 1, not -1, so we need to switch the signs. The correct numbers are -11 and 12. So we can factor the equation as: (x - 11)(x + 12) = 0 Now, we set each factor equal to zero and solve for x: x - 11 = 0 or x + 12 = 0 x = 11 or x = -12 Therefore, the number could be either 11 or -12. Both of these numbers satisfy the original condition that the sum of the number and its square is boxed{132} .