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 \( x^{2} \)

Let's analyze each part of the problem step by step. ### Given: - \( 0 \leq x \leq 7 \) - \( 1 \leq y \leq 5 \) --- ### (i) **Largest Possible Value of \( xy \)** To maximize the product \( xy \): - **Maximize \( x \)**: The largest value \( x \) can take is **7**. - **Maximize \( y \)**: The largest value \( y \) can take is **5**. Thus, the largest possible value of \( xy \) is: \[ xy = 7 \times 5 = 35 \] **Answer:** The largest possible value of \( xy \) is **35**. --- ### (ii) **Least Possible Value of \( x - y \)** To minimize the expression \( x - y \): - **Minimize \( x \)**: The smallest value \( x \) can take is **0**. - **Maximize \( y \)**: The largest value \( y \) can take is **5**. Thus, the least possible value of \( x - y \) is: \[ x - y = 0 - 5 = -5 \] **Answer:** The least possible value of \( x - y \) is **-5**. --- ### (iii) **Least and Greatest Possible Values of \( x^2 \)** Since \( x \) is in the range \( 0 \leq x \leq 7 \): - **Least \( x^2 \)**: This occurs when \( x \) is at its minimum. \[ x^2 = 0^2 = 0 \] - **Greatest \( x^2 \)**: This occurs when \( x \) is at its maximum. \[ x^2 = 7^2 = 49 \] **Answer:** - The least possible value of \( x^2 \) is **0**. - The greatest possible value of \( x^2 \) is **49**. --- ### **Summary of Answers:** 1. **(i)** Largest \( xy = 35 \) 2. **(ii)** Least \( x - y = -5 \) 3. **(iii)** Least \( x^2 = 0 \) and greatest \( x^2 = 49 \)