1. The problem asks for the third term in the Taylor series expansion of a function $f(x)$ around the point $a$. 2. The Taylor series expansion of $f(x)$ about $a$ is given by: $$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$$ 3. The first term is $f(a)$, the second term is $f'(a)(x-a)$, and the third term is: $$\frac{f''(a)}{2}(x-a)^2$$ 4. This term comes from the formula for the $n$th term in the Taylor series: $$\frac{f^{(n)}(a)}{n!}(x-a)^n$$ where $n=2$ for the third term (since the series starts at $n=0$). 5. Among the options given: - (a) is incorrect because it multiplies by $(x-a)$ instead of $(x-a)^2$ and divides by 2 incorrectly. - (b) is incorrect because it lacks division by 2 and the square on $(x-a)$. - (c) is correct: $\frac{f''(a)}{2}(x-a)^2$ matches the third term. - (d) is incorrect because it lacks division by 2. Final answer: (c) $\frac{f''(a)}{2}(x-a)^2$