1. Given that \( 0 \leq x \leq 7 \) and \( 1 \leq y \leq 5 \), find the (i) largest possible value of \( x y \), (ii) least possible \( x-y \), (iii) least and greatest possible values of \( x^{2} \)

To solve the given problems, we will analyze each part step by step. (i) To find the largest possible value of \( x y \) given the constraints \( 0 \leq x \leq 7 \) and \( 1 \leq y \leq 5 \), we multiply the maximum values of \( x \) and \( y \). The maximum is reached when \( x = 7 \) and \( y = 5 \): \[ xy = 7 \times 5 = 35. \] Thus, the largest possible value of \( xy \) is \( 35 \). (ii) For the least possible value of \( x - y \), we want to maximize \( y \) (to decrease the overall value of \( x - y\)) and minimize \( x \). The extreme values occur when \( x = 0 \) and \( y = 5 \): \[ x - y = 0 - 5 = -5. \] Thus, the least possible value of \( x - y \) is \(-5\). (iii) To find the least and greatest possible values of \( x^{2} \), we need to evaluate the values of \( x \) at the endpoints of the interval \( [0, 7] \). The least value occurs when \( x = 0 \): \[ x^{2} = 0^{2} = 0. \] The greatest value occurs when \( x = 7 \): \[ x^{2} = 7^{2} = 49. \] Therefore, the least possible value of \( x^{2} \) is \( 0 \) and the greatest possible value is \( 49 \). **Final Answers:** (i) 35 (ii) -5 (iii) Least: 0, Greatest: 49