1. **Problem statement:** Given a function $f(x) = g(x) - r(x)$ with conditions at $x=2$: $g'(2) = r'(2)$ and $g''(2) > r''(2)$, determine the nature of $f$ at $x=2$. 2. **Recall the derivative rules:** - The first derivative of $f$ is $f'(x) = g'(x) - r'(x)$. - The second derivative of $f$ is $f''(x) = g''(x) - r''(x)$. 3. **Evaluate the first derivative at $x=2$:** $$f'(2) = g'(2) - r'(2) = 0$$ This means $x=2$ is a critical point of $f$. 4. **Evaluate the second derivative at $x=2$:** $$f''(2) = g''(2) - r''(2) > 0$$ Since $g''(2) > r''(2)$, their difference is positive. 5. **Interpretation:** - If $f'(2) = 0$ and $f''(2) > 0$, then $f$ has a local minimum at $x=2$. **Final answer:** The function $f$ has a local minimum value at $x=2$.