1. **State the problem:** Find the maximum value of the function $$g(x) = -\left|(-x)^2 - l^2 + ml + z\right|$$ for real numbers $l$, $m$, and $z$. 2. **Rewrite the function:** Note that $(-x)^2 = x^2$, so $$g(x) = -\left|x^2 - l^2 + ml + z\right|.$$ 3. **Analyze the absolute value:** The absolute value $|A|$ is always non-negative, so $-|A| \leq 0$ for any real $A$. 4. **Maximum value of $g(x)$:** Since $g(x)$ is the negative of an absolute value, its maximum value is 0, which occurs when the expression inside the absolute value is zero: $$x^2 - l^2 + ml + z = 0.$$ 5. **Conclusion:** The maximum value of $g(x)$ is $$\boxed{0}.$$ --- **Question 45:** Which data set has the smallest standard deviation? - Standard deviation measures how spread out numbers are from the mean. - A set with all identical numbers has zero standard deviation. Data sets: - A: 0, 1, 2, 3, 4, 5 - B: 1, 1, 2, 2, 3, 3 - C: 4, 6, 8, 8, 12, 14 - D: 7, 7, 7, 7, 7, 7 Since set D has all identical values, its standard deviation is 0, which is the smallest possible. **Final answers:** - Max value of $g(x)$ is $0$. - Smallest standard deviation is from set D.