The function `f(x)=(log)_e(x^3+sqrt(x^6+1))` is of the following types: (a) even and increasing (b) odd and increasing (c) even and decreasing (d) odd and decreasing

Explanation

To determine the type of the function \( f(x) = \log_e(x^3 + \sqrt{x^6 + 1}) \), we need to analyze its derivative and check if it is odd or even.

### Step 1: Find the derivative \( f'(x) \)

Using the chain rule and the properties of logarithms, we differentiate \( f(x) \):

\[
f'(x) = \frac{1}{x^3 + \sqrt{x^6 + 1}} \cdot \frac{d}{dx}(x^3 + \sqrt{x^6 + 1})
...