Optimization — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

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

1. 1.

   Find the x-value where the function has its maximum.

   $$P(x)=-x^2+8x$$

   Answer: $4$

   Step-by-step solution

   1. $$P^{\prime }(x)=8-2x=0$$
   2. $$x=4,P^{\prime \prime }(x)=-2\text{ confirms a maximum.}$$
2. 2.

   Find the x-value where the function has its maximum.

   $$P(x)=-2x^2+12x$$

   Answer: $3$

   Step-by-step solution

   1. $$P^{\prime }(x)=12-4x=0$$
   2. $$x=3,P^{\prime \prime }(x)=-4\text{ confirms a maximum.}$$
3. 3.

   Find the x-value where the function has its minimum.

   $$P(x)=x^2-10x+30$$

   Answer: $5$

   Step-by-step solution

   1. $$P^{\prime }(x)=2x-10=0$$
   2. $$x=5,P^{\prime \prime }(x)=2\text{ confirms a minimum.}$$
4. 4.

   Find the x-value where the function has its minimum.

   $$P(x)=3x^2-18x+40$$

   Answer: $3$

   Step-by-step solution

   1. $$P^{\prime }(x)=6x-18=0$$
   2. $$x=3,P^{\prime \prime }(x)=6\text{ confirms a minimum.}$$
5. 5.

   Find the x-value where the function has its maximum.

   $$P(x)=-x^2+14x-24$$

   Answer: $7$

   Step-by-step solution

   1. $$P^{\prime }(x)=14-2x=0$$
   2. $$x=7,P^{\prime \prime }(x)=-2\text{ confirms a maximum.}$$
6. 6.

   Find the x-value where the function has its minimum.

   $$P(x)=2x^2-16x+35$$

   Answer: $4$

   Step-by-step solution

   1. $$P^{\prime }(x)=4x-16=0$$
   2. $$x=4,P^{\prime \prime }(x)=4\text{ confirms a minimum.}$$
7. 7.

   Find the x-value where the function has its maximum.

   $$P(x)=-x^2+6x$$

   Answer: $3$

   Step-by-step solution

   1. $$P^{\prime }(x)=6-2x=0$$
   2. $$x=3,P^{\prime \prime }(x)=-2\text{ confirms a maximum.}$$
8. 8.

   Find the x-value where the function has its maximum.

   $$P(x)=-3x^2+18x$$

   Answer: $3$

   Step-by-step solution

   1. $$P^{\prime }(x)=18-6x=0$$
   2. $$x=3,P^{\prime \prime }(x)=-6\text{ confirms a maximum.}$$
9. 9.

   Find the x-value where the function has its minimum.

   $$P(x)=x^2-4x+9$$

   Answer: $2$

   Step-by-step solution

   1. $$P^{\prime }(x)=2x-4=0$$
   2. $$x=2,P^{\prime \prime }(x)=2\text{ confirms a minimum.}$$
10. 10.

    Find the x-value where the function has its minimum.

    $$P(x)=2x^2-20x+7$$

    Answer: $5$

    Step-by-step solution

    1. $$P^{\prime }(x)=4x-20=0$$
    2. $$x=5,P^{\prime \prime }(x)=4\text{ confirms a minimum.}$$
11. 11.

    Find the x-value where the function has its maximum.

    $$P(x)=-4x^2+24x-5$$

    Answer: $3$

    Step-by-step solution

    1. $$P^{\prime }(x)=24-8x=0$$
    2. $$x=3,P^{\prime \prime }(x)=-8\text{ confirms a maximum.}$$
12. 12.

    Find the x-value where the function has its minimum.

    $$P(x)=5x^2-30x+11$$

    Answer: $3$

    Step-by-step solution

    1. $$P^{\prime }(x)=10x-30=0$$
    2. $$x=3,P^{\prime \prime }(x)=10\text{ confirms a minimum.}$$
13. 13.

    Find the x-value where the function has its maximum.

    $$P(x)=-2x^2+20x-13$$

    Answer: $5$

    Step-by-step solution

    1. $$P^{\prime }(x)=20-4x=0$$
    2. $$x=5,P^{\prime \prime }(x)=-4\text{ confirms a maximum.}$$
14. 14.

    Find the x-value where the function has its minimum.

    $$P(x)=3x^2-24x+50$$

    Answer: $4$

    Step-by-step solution

    1. $$P^{\prime }(x)=6x-24=0$$
    2. $$x=4,P^{\prime \prime }(x)=6\text{ confirms a minimum.}$$
15. 15.

    Find the x-value where the function has its maximum.

    $$P(x)=-5x^2+40x-17$$

    Answer: $4$

    Step-by-step solution

    1. $$P^{\prime }(x)=40-10x=0$$
    2. $$x=4,P^{\prime \prime }(x)=-10\text{ confirms a maximum.}$$
16. 16.

    Find the x-value where the function has its minimum.

    $$P(x)=4x^2-32x+19$$

    Answer: $4$

    Step-by-step solution

    1. $$P^{\prime }(x)=8x-32=0$$
    2. $$x=4,P^{\prime \prime }(x)=8\text{ confirms a minimum.}$$

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
• Curve Analysis
• Antiderivatives
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution