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Unit 6: Applications of Differentiation II — Extrema, Shape, and Optimization

Where a function peaks and dips, how its graph bends, the theorems guaranteeing those peaks exist, and turning a real constraint into the best answer.

17 topics · 64 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

4.3 Critical Points

  1. How it worksHow do you find a function's critical points by solving f'(x) = 0 and checking where f'(x) is undefined?
  2. Worked exampleFind the critical points of f(x) = x3 - 3x2 - 9x + 5.
  3. Common mistakeWhy isn't every critical point automatically a maximum or a minimum?
Your turn1 of 2
For f(x) = x3 - 6x2 - 13x + 8, find the sum of its critical points.

4.3 The Extreme Value Theorem

  1. How it worksHow does chopping an interval into smaller and smaller pieces prove a continuous function must reach a maximum?
  2. Going furtherWhat happens to the Extreme Value Theorem's guarantee when the interval is open instead of closed?
  3. Common mistakeWhy doesn't the Extreme Value Theorem guarantee a maximum for f(x) = 1/x on the open interval from 0 to 1?
Your turn1 of 2
Consider f(x) = 6/x on the interval (0, 6] — closed on the right, but open on the left at 0. Does the Extreme Value Theorem guarantee f has an absolute maximum on this interval?

4.3 The Candidates Test for Absolute Extrema

  1. How it worksHow does comparing a function's value at its critical points and at both endpoints locate its absolute extrema?
  2. Common mistakeWhy isn't checking only the critical points enough to find an absolute maximum on a closed interval?
  3. Worked exampleFind the absolute maximum and minimum of f(x) = x3 - 6x2 + 5 on the interval from -1 to 5.
  4. Worked exampleFind the absolute maximum and minimum of f(x) = x - 2sin(x) on the interval from 0 to 2·π.
Your turn1 of 2
Find the absolute maximum of f(x) = x3 - 4x2 on the interval [0, 4 + 8].

4.5 Increasing and Decreasing Intervals from the First Derivative

  1. How it worksHow do you use a sign chart of f'(x) to mark off exactly where f(x) increases and decreases?
  2. Common mistakeWhy doesn't a negative function value mean the function itself is decreasing at that point?
  3. Worked exampleFind the intervals where f(x) = x3 - 3x2 is increasing and decreasing.
Your turn1 of 2
For f(x) = 4x3 - 7x, find the right endpoint of the interval on which f is decreasing.

4.5 The First Derivative Test

  1. How it worksHow does checking the sign of f' just before and just after a critical point classify it as a max, min, or neither?
  2. Worked exampleFind and classify the critical points of f(x) = x4 - 4x3 using the First Derivative Test.
  3. Common mistakeWhy doesn't every critical point where the first derivative test is inconclusive fail to be an extremum?
  4. Worked exampleFind and classify the critical points of f(x) = x3 - 12x using the First Derivative Test.
  5. Going furtherWhat happens to the First Derivative Test's conclusion when the first derivative doesn't change sign at all across a critical point?
Your turn1 of 2
For f(x) = x3 - 7x, use the First Derivative Test to find the x-coordinate of the local minimum.

4.5 Concavity and the Second Derivative

  1. Going furtherWhat happens to a function's concavity at a point where its second derivative is undefined but the function itself is still continuous?
  2. How it worksHow does the sign of the second derivative decide whether a graph bends upward or downward?
  3. Common mistakeWhy isn't a decreasing function automatically concave down?
  4. Worked exampleFind the intervals of concavity for f(x) = x4 - 6x2.
Your turn1 of 2
For f(x) = 4x4 - 7x2, find the right endpoint of the interval on which f is concave down.

4.5 Inflection Points

  1. How it worksHow do you find an inflection point by solving f''(x) = 0 and checking that concavity actually changes there?
  2. Common mistakeWhy doesn't every solution to f''(x) = 0 automatically become an inflection point?
  3. Worked exampleFind the inflection points of f(x) = x3 - 6x2 + 9x + 1.
Your turn1 of 2
For f(x) = x3 - 7x2 + 6x + 8, find the x-coordinate of the inflection point.

4.5 The Second Derivative Test

  1. How it worksHow does a positive or negative second derivative at a critical point classify it as a local min or max?
  2. Worked exampleFind and classify the critical points of f(x) = x3 - 3x using the Second Derivative Test.
  3. Common mistakeWhy doesn't the Second Derivative Test decide anything when the second derivative comes out to exactly zero?
Your turn1 of 2
For f(x) = x3 - 7x, find f''(x).

4.5 Sketching a Graph from Its Derivatives

  1. How it worksHow do intercepts, asymptotes, increasing and decreasing intervals, and concavity combine into one finished sketch?
  2. Common mistakeWhy doesn't a function need to cross the x-axis to have a well-defined shape you can sketch from its derivatives?
  3. Worked exampleSketch the graph of f(x) = x3 - 3x2 using its critical points, inflection point, and end behavior.
  4. Going furtherWhat happens to a curve sketch when the function has a vertical asymptote in the middle of the domain?
Your turn1 of 2
For f(x) = x3 - 7x2, find the y-coordinate of the inflection point, to help sketch the graph.

4.7 Setting Up an Optimization Problem

  1. How it worksHow do you turn a real-world goal and a constraint into a single-variable function ready to be optimized?
  2. Common mistakeWhy doesn't skipping the constraint equation and optimizing two variables directly still work?
  3. Worked exampleBuild the function to minimize for the surface area of a box with a square base and fixed volume of 32 cubic meters.
Your turn1 of 2
A box with a square base of side x and fixed volume 38 cubic meters has its height determined by the volume constraint. Write its surface area S(x) as a function of x alone, after eliminating the height.

4.7 Solving a Constrained Optimization Problem

  1. How it worksHow does the Candidates Test finish an optimization problem once the objective is down to one variable?
  2. Worked exampleFind the point on the curve y = square root of x closest to the point (4, 0).
  3. Worked exampleFind the dimensions of the rectangle with perimeter 40 meters that has the maximum possible area.
  4. Common mistakeWhy doesn't a critical point in an optimization problem automatically give the best answer without checking it's a max?
Your turn1 of 2
Find the maximum possible area of a rectangle with perimeter 62 meters.

4.8 L'Hopital's Rule for 0/0 and infinity/infinity

  1. How it worksHow does replacing a numerator and denominator with their derivatives untangle a 0/0 or infinity/infinity limit?
  2. Worked exampleFind the limit of (sin x)/x as x approaches 0 using L'Hopital's Rule.
  3. Worked exampleFind the limit of (ex)/x2 as x approaches infinity using L'Hopital's Rule.
  4. Common mistakeWhy doesn't L'Hopital's Rule apply to a limit that is not already in the form 0/0 or infinity/infinity?
  5. Worked exampleFind the limit of (x - sin x)/x3 as x approaches 0 using L'Hopital's Rule twice.
Your turn1 of 2
Find the limit of (4x - sin(4x))/x3 as x approaches 0 using L'Hopital's Rule (applied twice).

4.8 L'Hopital's Rule for 0 times infinity and infinity minus infinity

  1. How it worksHow do you rewrite a 0 times infinity or infinity minus infinity limit as a single fraction so L'Hopital's Rule applies?
  2. Worked exampleFind the limit of x·ln(x) as x approaches 0 from the right by rewriting it as a quotient.
  3. Worked exampleFind the limit of (1/x - 1/sin x) as x approaches 0 by combining the fractions first.
Your turn1 of 2
Find the limit of 4·x·ln(x) + 6 as x approaches 0 from the right, by rewriting the product as a quotient.

4.8 L'Hopital's Rule for the Indeterminate Powers

  1. How it worksHow does taking a logarithm turn an indeterminate power like 00 or 1infinity into a product L'Hopital's Rule can handle?
  2. Worked exampleFind the limit of (1 + 1/x)x as x approaches infinity using logarithms and L'Hopital's Rule.
  3. Common mistakeWhy isn't the final answer to an indeterminate power problem just whatever value the logarithm limit comes out to?
  4. Worked exampleFind the limit of xx as x approaches 0 from the right using logarithms and L'Hopital's Rule.
Your turn1 of 2
Find the limit of (1 + 4/x)x as x approaches infinity, using logarithms and L'Hopital's Rule.

4.8 Comparing Growth Rates: Exponentials, Polynomials, and Logarithms

  1. How it worksHow does applying L'Hopital's Rule over and over eventually prove ex beats every power of x?
  2. Worked exampleFind the limit of x10/ex as x approaches infinity using repeated L'Hopital's Rule.
  3. Worked exampleFind the limit of ln(x)/x as x approaches infinity using L'Hopital's Rule.
  4. Common mistakeWhy isn't x100 eventually bigger than 2x for all large x, just because 100 is a huge exponent?
  5. Going furtherWhat happens to the ranking of growth rates when you compare ln(x), x, and x2 all at once as x approaches infinity?
Your turn1 of 2
Find the limit of 4·x7/ex + 8 as x approaches infinity, using repeated L'Hopital's Rule on the first term.

4.9 Newton's Method for Approximating Roots

  1. How it worksWatch each tangent line in Newton's Method land closer to the true root than the guess before it.
  2. Worked exampleFind one iteration of Newton's Method starting at x = 2 toward the root of f(x) = x2 - 5.
  3. Worked exampleFind two iterations of Newton's Method starting at x = 1 toward the root of f(x) = x3 - 2.
  4. Common mistakeWhy doesn't Newton's Method give the exact root after just one or two iterations in general?
  5. Ties togetherHow does Newton's Method use the same tangent-line idea that powers linear approximation, just applied over and over?
Your turn1 of 2
Using Newton's Method, find one iteration starting at x0 = 4 toward the root of f(x) = x2 - 7.

4.9 When Newton's Method Fails

  1. How it worksHow does a nearly flat tangent line at a starting guess catapult the next guess far away from the root?
  2. Worked exampleShow what happens when Newton's Method is applied to f(x) = x1/3 starting anywhere other than the root.
  3. Going furtherWhat happens to Newton's Method when it lands exactly on a critical point of the function?
Your turn1 of 2
For f(x) = x1/3, Newton's Method's next iterate from x0 = 6 is x1 = x0 - f(x0)/f'(x0). Find x1.