1. **Problem Statement:** We have a smooth curve $f(x)$ defined on $(-\infty, \infty)$ with: - A local maximum at $x=a$ - A local minimum at $x=b$ - A point of concavity change (inflection point) at $x=c$ We need to determine the signs of $f'(a)$, $f''(a)$, $f'(b)$, $f''(b)$, $f'(c)$, and $f''(c)$ and predict the shape of the curve. 2. **Key Formulas and Rules:** - At a local maximum or minimum, the first derivative $f'(x)$ is zero. - The second derivative $f''(x)$ tells us about concavity: - $f''(x) > 0$ means concave up. - $f''(x) < 0$ means concave down. - At an inflection point, $f''(x) = 0$ and the concavity changes. 3. **Analyzing the points:** - At $x=a$ (local maximum): - $f'(a) = 0$ because slope is zero at maxima. - $f''(a) < 0$ because the curve is concave down at a maximum. - At $x=b$ (local minimum): - $f'(b) = 0$ because slope is zero at minima. - $f''(b) > 0$ because the curve is concave up at a minimum. - At $x=c$ (inflection point): - $f''(c) = 0$ because concavity changes here. - $f'(c)$ can be positive, negative, or zero depending on the function; it is not necessarily zero. 4. **Shape Prediction:** - For $x < c$, the curve is concave up. - For $x > c$, the curve is concave down. - The curve has a peak at $x=a$ and a trough at $x=b$. - Label points on the $x$-axis as $a$, $b$, and $c$ accordingly. 5. **Exploring Inflection Points and Stationary Inflection Points:** - An inflection point is where $f''(x)$ changes sign. - At these points, the curve changes concavity and the rate of change of slope is maximal. - A stationary inflection point is an inflection point where $f'(x) = 0$ as well. **Examples:** - $f(x) = x^3$ has an inflection point at $x=0$ with $f'(0) = 0$ (stationary inflection point). - $f(x) = x^3 + x$ has an inflection point at $x=0$ but $f'(0) = 1 \neq 0$ (non-stationary inflection point). This distinction helps understand the behavior of the curve at inflection points.