Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.

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1. Step 1: Understand the problem. We need to find a value of 'a' such that the derivative of the piecewise function f(x) is continuous at x=1. The function is defined as f(x) = 2x^2 for x ≤ 1 and f(x) = ax - 2 for x > 1.
2. Step 2: Ensure f(x) is continuous at x=1. For f(x) to be continuous at x=1, the left-hand limit (as x approaches 1 from the left) must equal the right-hand limit (as x approaches 1 from the right) and also equal f(1). Calculate these limits and set them equal to each other.
3. Step 3: Differentiate each piece of the function. Find f'(x) for x ≤ 1 and for x > 1. For x ≤ 1, differentiate 2x^2 to get f'(x) = 4x. For x > 1, differentiate ax - 2 to get f'(x) = a.
4. Step 4: Ensure f'(x) is continuous at x=1. For f'(x) to be continuous at x=1, the left-hand derivative (as x approaches 1 from the left) must equal the right-hand derivative (as x approaches 1 from the right). Set 4(1) equal to a.
5. Step 5: Solve for 'a'. From the equation 4 = a, determine the value of 'a' that makes f'(x) continuous at x=1.