Concavity and inflection points help us understand how functions curve and change direction. They're key to grasping a function's shape and behavior, which is crucial for solving real-world problems in calculus.

By analyzing a function's second derivative, we can determine where it's curving upward or downward. This knowledge is super useful for finding maximum and minimum points, and for understanding how quickly a function is increasing or decreasing.

Concavity and Inflection Points

Definition of concavity and inflection points

• Concavity describes the curvature of a function's graph
  • Concave up graph curves upward like a cup (parabola $y=x^2$ for $x>0$)
  • Concave down graph curves downward like a dome (parabola $y=-x^2$)
• Concavity determined by the second derivative $f''(x)$
  • $f''(x) > 0$ on an interval indicates concave up (exponential function $y=e^x$)
  • $f''(x) < 0$ on an interval indicates concave down (logarithmic function $y=\ln x$)
• Inflection point is where the concavity changes from up to down or down to up
  • $f''(x) = 0$ or undefined at an inflection point (cubic function $y=x^3$ at $x=0$)

Intervals of concave up vs down

• Calculate the second derivative $f''(x)$ of the function
• Set $f''(x) = 0$ and solve for $x$ to find potential inflection points
• Analyze the sign of $f''(x)$ on intervals between potential inflection points
  • $f''(x) > 0$ on an interval means concave up (quadratic function $y=x^2+1$ for all $x$)
  • $f''(x) < 0$ on an interval means concave down (quadratic function $y=-x^2+1$ for all $x$)
• Construct a sign chart or number line to visualize concavity intervals (mark $+$ for concave up, $-$ for concave down)

Identification of inflection points

• Potential inflection points occur where $f''(x) = 0$ or is undefined
• Evaluate points on either side of a potential inflection point
  • Inflection point confirmed if concavity changes (cubic function $y=x^3$ at $x=0$)
  • Not an inflection point if concavity remains the same (quadratic function $y=x^2$ at $x=0$)
• Check that the function is continuous at the inflection point (cubic function $y=x^3$ is continuous at $x=0$)

Concavity and function behavior

• Concave up functions increase at an increasing rate (accelerating)
  • Tangent lines fall below the graph (exponential function $y=2^x$)
• Concave down functions increase at a decreasing rate or decrease at an increasing rate (decelerating)
  • Tangent lines sit above the graph (square root function $y=\sqrt{x}$)
• At an inflection point, the rate of change switches between accelerating and decelerating
  • Second derivative $f''(x)$ changes sign (sine function $y=\sin x$ at multiples of $\pi$)

Concavity in optimization problems

1. Identify the objective function to optimize (e.g., profit, cost, area)
2. Calculate the first and second derivatives of the objective function
3. Determine the concavity of the objective function

• Concave up critical point is a local minimum (quadratic function $y=x^2$ at $x=0$)
• Concave down critical point is a local maximum (quadratic function $y=-x^2$ at $x=0$)

1. Find the critical points by setting the first derivative equal to zero
2. Evaluate the objective function at the critical points to find the optimal solution
3. Interpret the results in the context of the problem (e.g., maximum profit, minimum cost)