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Critical Points Practice Problems

Find f′(x), solve f′(x) = 0, and find where f′(x) is undefined. Keep only values in the domain of f, and list every critical point separated by commas.

  1. 1.

    Find the critical point of .

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    gives .

  2. 2.

    Find all critical points of . Separate answers with commas.

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    gives and .

  3. 3.

    Find all critical points of .

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    , so and .

  4. 4.

    Find all critical points of .

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    , so , and .

  5. 5.

    Find the critical point of .

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    is never 0 and is undefined at , where is defined. So .

  6. 6.

    Find the critical point of .

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    . Since , the only critical point is .

  7. 7.

    Find all critical points of .

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    . The numerator is 0 at , and the denominator is never 0.

  8. 8.

    Find all critical points of .

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    . The numerator gives . The derivative is undefined at , but 0 is not in the domain of , so it is not a critical point.

  9. 9.

    Find all critical points of on . You can type pi, for example pi/3.

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    gives , so and .

  10. 10.

    Find all critical points of .

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    , so . It is 0 at and undefined at , where is defined.

  11. 11.

    Find the critical point of . You can type e, for example 1/e.

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    , so and .

  12. 12.

    Find all critical points of .

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    Between and , , so at . At the graph has corners, so is undefined there, and is defined. The critical points are , and .
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Critical Points Practice Problems

These problems follow the critical points lesson. They start with polynomials and build up to roots, rational, trig and exponential functions, and points where the derivative is undefined.

The method

  1. Find the domain of .
  2. Find and write it as one factored fraction.
  3. Solve : set the numerator to 0.
  4. Find where is undefined: set the denominator to 0.
  5. Keep only the values that are in the domain of .

Common mistakes to watch for

  • Missing points where is undefined, such as cusps and corners.
  • Including values outside the domain, such as vertical asymptotes.
  • Dividing by a factor like instead of factoring, which loses the solution .

Frequently Asked Questions

How should I type my answer?+

List the x-values separated by commas, in any order, for example -2, 0, 2. You can type pi and e, such as pi/3 or 1/e.

Do I need to include points where the derivative is undefined?+

Yes, if the function itself is defined there. Points outside the domain of f, such as x = 0 for 9/x, are never critical points.

More practice problems

Power Rule Practice Problems Quotient Rule Practice Problems Higher-Order Derivatives Practice Problems