1. Use the intermediate value theorem to prove that the equation x = cosx has at least one real solution. Using the Intermediate Value Theorem We want to prove that the equation zd = cos(m) has at least one real solution using the Intermediate Value Theorem (IVT) 1. Function definition: Define the function f(z) = z* cos(z). 2. Check continuity: Both and cos(2) are continuous functions for all real numbers. Therefore, their difference f(z) = 23 cos(z) is also continuous for all real numbers. 3. Choose interval: We need to select an interval where f(x) is continuous and where we can find two points with opposite signs to apply the IVT. 4. Select a and b: Let's choose g = Q0 and b = % These values are convenient points where we can easily evaluate the function. 5. Calculate f(a) and f(b): (0) = 0* cos(0) = 1 1(3)- (@) -()-F0-% 6. Determine signs of f(a) and f(b): f(0) > 0 (positive) f (%) >0 (positive) 7. Apply the Intermediate Value Theorem (IVT): Since f(a) and f(b) have opposite signs (one positive and one positive), and f(z) is continuous on [0, %, the IVT guarantees the existence of at least one c in (0, ) such that f(c) = 0. This means there is at least one real solution to the equation z = cos(z). 3 Conclusion: Therefore, the equation = cos(z) has at least one real solution in the interval (0, 3)