Curve Analysis — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the critical x-value.

   $$f(x)=x^2-4x$$

   Answer: $2$

   Step-by-step solution

   1. $$f^{\prime }(x)=2x-4$$
   2. $$f^{\prime }(x)=0\Rightarrow x=2$$
2. 2.

   Find the critical x-value.

   $$f(x)=x^2+6x$$

   Answer: $-3$

   Step-by-step solution

   1. $$f^{\prime }(x)=2x+6$$
   2. $$f^{\prime }(x)=0\Rightarrow x=-3$$
3. 3.

   Find the x-coordinate of the inflection point.

   $$f(x)=x^3-3x^2$$

   Answer: $1$

   Step-by-step solution

   1. $$f^{\prime \prime }(x)=6\left(x-1\right)$$
   2. $$f^{\prime \prime }(x)=0\Rightarrow x=1$$
4. 4.

   Find the x-coordinate of the inflection point.

   $$f(x)=x^3+6x^2$$

   Answer: $-2$

   Step-by-step solution

   1. $$f^{\prime \prime }(x)=6\left(x+2\right)$$
   2. $$f^{\prime \prime }(x)=0\Rightarrow x=-2$$
5. 5.

   Find the positive critical x-value.

   $$f(x)=x^4-8x^2$$

   Answer: $2$

   Step-by-step solution

   1. $$f^{\prime }(x)=4x^3-16x$$
   2. $$f^{\prime }(x)=0\Rightarrow x=2$$
6. 6.

   Find the positive critical x-value.

   $$f(x)=x^3-12x$$

   Answer: $2$

   Step-by-step solution

   1. $$f^{\prime }(x)=3x^2-12$$
   2. $$f^{\prime }(x)=0\Rightarrow x=2$$
7. 7.

   Find the critical x-value.

   $$f(x)=x^2-6x$$

   Answer: $3$

   Step-by-step solution

   1. $$f^{\prime }(x)=2x-6$$
   2. $$f^{\prime }(x)=0\Rightarrow x=3$$
8. 8.

   Find the critical x-value.

   $$f(x)=x^2+10x$$

   Answer: $-5$

   Step-by-step solution

   1. $$f^{\prime }(x)=2x+10$$
   2. $$f^{\prime }(x)=0\Rightarrow x=-5$$
9. 9.

   Find the critical x-value.

   $$f(x)=x^2-2x$$

   Answer: $1$

   Step-by-step solution

   1. $$f^{\prime }(x)=2x-2$$
   2. $$f^{\prime }(x)=0\Rightarrow x=1$$
10. 10.

    Find the x-coordinate of the inflection point.

    $$f(x)=x^3-6x^2$$

    Answer: $2$

    Step-by-step solution

    1. $$f^{\prime \prime }(x)=6\left(x-2\right)$$
    2. $$f^{\prime \prime }(x)=0\Rightarrow x=2$$
11. 11.

    Find the x-coordinate of the inflection point.

    $$f(x)=x^3+9x^2$$

    Answer: $-3$

    Step-by-step solution

    1. $$f^{\prime \prime }(x)=6\left(x+3\right)$$
    2. $$f^{\prime \prime }(x)=0\Rightarrow x=-3$$
12. 12.

    Find the x-coordinate of the inflection point.

    $$f(x)=x^3-15x^2$$

    Answer: $5$

    Step-by-step solution

    1. $$f^{\prime \prime }(x)=6\left(x-5\right)$$
    2. $$f^{\prime \prime }(x)=0\Rightarrow x=5$$
13. 13.

    Find the positive critical x-value.

    $$f(x)=x^4-18x^2$$

    Answer: $3$

    Step-by-step solution

    1. $$f^{\prime }(x)=4x^3-36x$$
    2. $$f^{\prime }(x)=0\Rightarrow x=3$$
14. 14.

    Find the positive critical x-value.

    $$f(x)=x^3-27x$$

    Answer: $3$

    Step-by-step solution

    1. $$f^{\prime }(x)=3x^2-27$$
    2. $$f^{\prime }(x)=0\Rightarrow x=3$$
15. 15.

    Find the positive critical x-value.

    $$f(x)=x^4-32x^2$$

    Answer: $4$

    Step-by-step solution

    1. $$f^{\prime }(x)=4x^3-64x$$
    2. $$f^{\prime }(x)=0\Rightarrow x=4$$
16. 16.

    Find the x-coordinate of the inflection point.

    $$f(x)=x^3+12x^2$$

    Answer: $-4$

    Step-by-step solution

    1. $$f^{\prime \prime }(x)=6\left(x+4\right)$$
    2. $$f^{\prime \prime }(x)=0\Rightarrow x=-4$$

More from Calculus I

• Evaluating Limits
• One-Sided Limits
• Limits at Infinity
• Continuity
• Derivative from First Principles
• Power Rule
• Trigonometric Derivatives
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Related Rates
• Optimization
• Antiderivatives
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution