Calculus I: Critical Points, Extrema, Optimization, and Indeterminate Forms

Absolute Maxima and Minima

Finding Absolute Extrema

Absolute extrema refer to the highest and lowest values a function attains on a given interval. To find these, evaluate the function at all critical points and endpoints within the interval.

• Critical Point: A point where the derivative is zero or undefined.

• Endpoint: The boundary values of the interval.

• Procedure:

  1. Find the derivative of the function.

  2. Solve for critical points by setting the derivative equal to zero or where it is undefined.

  3. Evaluate the function at all critical points and endpoints.

  4. Compare values to determine absolute maximum and minimum.

• Example: For $f(x) = \frac{1}{2}x^2 - 5$ on $-2 \leq x \leq 3$, find $f'(x) = x$. Critical point at $x=0$. Evaluate $f(-2)$, $f(0)$, $f(3)$ to find absolute max/min.

Critical Points, Increasing, and Decreasing

Identifying Critical Points

Critical points occur where the first derivative of a function is zero or undefined. These points are candidates for local maxima, minima, or points of inflection.

• Increasing Interval: Where $f'(x) > 0$.

• Decreasing Interval: Where $f'(x) < 0$.

• Local Maximum: If $f'(x)$ changes from positive to negative at a critical point.

• Local Minimum: If $f'(x)$ changes from negative to positive at a critical point.

• Example: For $f(x) = -2x^2 - 2x^3$, $f'(x) = -4x - 6x^2$. Set $f'(x) = 0$ to find critical points, then test intervals to determine increasing/decreasing behavior.

Points of Inflection and Concavity

Second Derivative Test

Points of inflection occur where the concavity of a function changes, i.e., where the second derivative changes sign. Concave up means the graph opens upward; concave down means it opens downward.

• Concave Up: Where $f''(x) > 0$.

• Concave Down: Where $f''(x) < 0$.

• Point of Inflection: Where $f''(x) = 0$ and the sign of $f''(x)$ changes.

• Procedure:

  1. Find $f''(x)$.

  2. Solve $f''(x) = 0$ for possible inflection points.

  3. Test intervals around these points to determine concavity.

• Example: For $f(x) = x^3 - 2x^2 + 3$, $f''(x) = 6x - 4$. Set $f''(x) = 0$ to find inflection points.

Optimization

Solving Optimization Problems

Optimization involves finding the maximum or minimum value of a function subject to certain constraints. This is commonly used in real-world applications such as maximizing area or minimizing cost.

• Steps:

  1. Express the quantity to be optimized as a function of one variable.

  2. Identify constraints and use them to reduce the number of variables.

  3. Find the derivative and solve for critical points.

  4. Test critical points and endpoints to find the optimal value.

• Example: To minimize the product of two integers whose sum is 12, let $x$ and $12-x$ be the integers. The product is $P = x(12-x)$. Find $P'(x)$, set to zero, and solve for $x$.

Indeterminate Forms and Limits

Evaluating Indeterminate Forms

Indeterminate forms arise in limits where direct substitution does not yield a clear answer, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. L'Hôpital's Rule is often used to resolve these forms.

• Common Indeterminate Forms: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$

• L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit exists).

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

HTML Table: Summary of Critical Point Classification

Type	Test	Result
Local Maximum	First Derivative changes from + to -	Peak point
Local Minimum	First Derivative changes from - to +	Valley point
Point of Inflection	Second Derivative changes sign	Concavity changes

Additional info:

• Some questions involve real-world applications, such as coffee cooling and aquarium design, which require setting up equations based on geometric or physical constraints.

• For trigonometric and polynomial functions, always check the domain and endpoints when finding extrema.