Step 1: Concept Overview:
For a differentiable function, a necessary condition for a local maximum or minimum at an interior point is that its first derivative equals zero at that point.
Step 2: Methodology:
1. Determine the function's first derivative, \(f'(x)\).
2. Given that the maximum on \([0, 2]\) occurs at \(x=1\), then \(f'(1) = 0\).
3. Solve \(f'(1) = 0\) to find 'a'.
Step 3: Detailed Solution:
Given \(f(x) = x^4 - 62x^2 + ax + 9\), we find the derivative:
\[ f'(x) = \frac{d}{dx}(x^4 - 62x^2 + ax + 9) \]\
\[ f'(x) = 4x^3 - 124x + a \]\
Since the maximum occurs at \(x=1\) within the interval \([0, 2]\), and \(x=1\) is an interior point, it's a critical point where the derivative is zero. Thus:
\[ f'(1) = 0 \]\
Substituting \(x=1\) into the derivative:
\[ 4(1)^3 - 124(1) + a = 0 \]\
\[ 4 - 124 + a = 0 \]\
\[ -120 + a = 0 \]\
\[ a = 120 \]\
Confirming it's a maximum using the second derivative test: \(f''(x) = 12x^2 - 124\). At \(x=1\), \(f''(1) = 12 - 124 = -112\), which is negative, indicating a local maximum.
Step 4: Solution:
The value of 'a' is 120.