<p><strong>Ex. 67</strong> The function \(f(x)\) is</p>
Step-by-Step Solution
Key Concept: If the derivative is an even function and $f(0) = 0$, then the original function is odd.
<p>Since $f'(t) = 3t^2 - 2t + 1 \geq 0$ for $t \in (0,1)$, $f(t)$ is increasing on $(0,1)$. Given $f(0) = 0$ and the symmetry of $f'(x)$ about the y-axis (even derivative), $f(x)$ must be an odd function.</p>
Correct Answer: b