Find possible values of 'a' such that f(x) `=e^(2x) -2(a^(2) -21) e^(x)+ 8x+5` is monotonically increasing for `x in R.`

Explanation

To find the possible values of 'a' such that the function \( f(x) = e^{2x} - 2(a^2 - 21)e^x + 8x + 5 \) is monotonically increasing for all \( x \in \mathbb{R} \), we need to follow these steps:

### Step 1: Find the first derivative of \( f(x) \)
The first derivative \( f'(x) \) is given by:
\[ f'(x) = \frac{d}{dx}(e^{2x}) - 2 \frac{d}{dx}((a^2 - 21)e^x) + \frac{d}{dx}(8x) + \frac{d}{dx}(5) \]
Calculating each term:
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