Suppose that x is the probability that a randomly selected person is left handed. The value (1-x) is the probability that the person is not left handed. In a sample of 1000 people, the function V(x)=1000x(1-x) represents the variance of the number of left-handed people in a group of 1000. What is the maximum variance?

Answer:The maximum variance is 250.Step-by-step explanation:Consider the provided function.[tex]V(x)=1000x(1-x)[/tex][tex]V(x)=1000x-1000x^2[/tex]Differentiate the above function as shown:[tex]V'(x)=1000-2000x[/tex]The double derivative of the provided function is:[tex]V''(x)=-2000[/tex]To find maximum variance set first derivative equal to 0.[tex]1000-2000x=0[/tex][tex]x=\frac{1}{2}[/tex]The double derivative of the function at [tex]x=\frac{1}{2}[/tex] is less than 0.Therefore, [tex]x=\frac{1}{2}[/tex] is a point of maximum.Thus the maximum variance is:[tex]V(x)=1000(\frac{1}{2})-1000{\frac{1}{2}}^2[/tex][tex]V(x)=250[/tex]Hence, the maximum variance is 250.