Graphs of Functions & Their Derivatives

Introduction to Concavity

#Understanding Concavity

Concavity describes the direction in which a curve bends and is related to the second derivative of a function.

#Concave Up and Concave Down

A curve is:

• Concave up if ${f}^{\text{'}\text{'}}(x) \ge 0$ for all values of $x$ in an interval.

  • ${f}^{\text{'}}(x)$ is increasing in this interval.
  • For example, the function $f(x) = x^2$ is concave up because ${f}^{\text{'}\text{'}}(x) = 2 \ge 0$.

• Concave down if ${f}^{\text{'}\text{'}}(x) \le 0$ for all values of $x$ in an interval.

  • ${f}^{\text{'}}(x)$ is decreasing in this interval.
  • For example, the function $f(x) = -x^2$ is concave down because ${f}^{\text{'}\text{'}}(x) = -2 \le 0$.

#Points of Inflection

A point of inflection is where a graph changes concavity, from concave up to concave down or vice versa.

• At a point of inflection, the second derivative is zero: ${f}^{\text{'}\text{'}}(x) = 0$.
• However, not every point where ${f}^{\text{'}\text{'}}(x) = 0$ is a point of inflection. The concavity must change.

For example, consider $f(x) = x^4$:

• ${f}^{\text{'}\text{'}}(x) = 12x^2$
• ${f}^{\text{'}\text{'}}(0) = 0$, but $x = 0$ is not a point of inflection because ${f}^{\text{'}\text{'}}(x) = 12x^2 > 0$ for all $x \ne 0$.

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#Exam Tips

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#Worked Example

The function $f$ is defined by:

$$f(x) = \sin x, \quad 0 \le x \le 2\pi$$

State the interval for which $f$ is concave down.

Solution:

A function is concave down when ${f}^{\text{'}\text{'}}(x)$ is negative.

$$\begin{align*} {f}^{\text{'}}(x) &= \cos x \\ {f}^{\text{'}\text{'}}(x) &= -\sin x \end{align*}$$

We need to solve: $$-\sin x \le 0, \quad 0 \le x \le 2\pi$$

The function is concave down where $-\sin x \le 0$.

By inspecting the graph of $y = -\sin x$ for $0 \le x \le 2\pi$, we see that the graph is less than zero on the interval $[0, \pi]$.

Thus, the function $f(x)$ is concave down on the interval $[0, \pi]$.

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#Practice Questions

Practice Question

1. Determine the concavity of the function $f(x) = x^3 - 3x^2 + 2$.

Practice Question

2. Find the points of inflection for the function $f(x) = \frac{1}{3}x^3 - x^2 + 2x - 1$.

Practice Question

3. For the function $f(x) = e^x$, identify the intervals where the function is concave up and concave down.

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#Glossary

• Concavity: The direction in which a curve bends.
• Second Derivative: The derivative of the first derivative of a function, denoted ${f}^{\text{'}\text{'}}(x)$.
• Concave Up: When ${f}^{\text{'}\text{'}}(x) \ge 0$ in an interval.
• Concave Down: When ${f}^{\text{'}\text{'}}(x) \le 0$ in an interval.
• Point of Inflection: A point where the graph changes concavity.

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#Summary and Key Takeaways

• Concavity is determined by the second derivative of a function.
• A function is concave up if ${f}^{\text{'}\text{'}}(x) \ge 0$ and concave down if ${f}^{\text{'}\text{'}}(x) \le 0$.
• Points of inflection occur where the second derivative is zero, and the concavity changes.
• Use visual aids like graphs to better understand concavity and points of inflection.

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#Real-World Applications

Concavity is crucial in various fields such as economics, physics, and engineering. For instance:

• In economics, concavity can help determine the increasing or decreasing returns of a production function.
• In physics, it helps in understanding the motion of objects under various forces.
• In engineering, it aids in the design of curves and structures to ensure stability and efficiency.

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Remember to practice these concepts regularly and apply them to different problems to master the topic of concavity!