Show that the variance of a binomial distribution with parameters n and p cannot exceed n/4.
Maximum value! NOT minimum value. Draw the graph of over the domain . (Did you bother to do that before just plucking a random [wrong] answer out of the air?). The maximum value occurs at the turning point. Drawing parabolas and finding turning points is something you should have learned how to do long ago. Do you realise that work previously studied is assumed to be understood?
I believe that the maximum variance for a binomial distribution occurs when the probability is 1/2 and the minimum variance occurs when probably is 0 or 1. The function of variance is in fact an inverted parabola however the maximal value occurs at 1/2 not 0. A quick check is to plot the variance over different value of p in the range of 0 and 1. A better way is to calculate first derivative = 0 and solve the for p which shows the apex of the parabola --- in this case p is also 1/2.