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Calculus Polynomial Graph Exercises

Exercises from public domain textbooks. Solutions where provided do not necessary follow the same lines as the original.

Solutions are not given for exercises whose number is not linked

Some workings are given for exercises which do not have a textbook solution. These may contain errors.

Graphs of polynomials: maxima and minima

Sketch the graphs. Find where the curves cut the axes. Describe y as x gets large. Find minima and maxima and points of inflections. Draw the tangent line at each point of inflection. (No textbook answers.)

D33-1. {$$ y = x^3 - 6x^2 + 9x + 3 $$} worked

D33-2. {$$ y = 4 + 3x - x^3 $$} worked

D33-3. {$$ y = x^3 - 3x^2 + 6x +10 $$} worked

D33-4. {$$ y = (x-3)^2(x-2) $$}


D33-5. {$$ y = (1-x^2)^3 $$}

worked Warning! The working does not agree with the textbook answer.

D33-6. {$$ y = (4-x^2)^2 $$}


D33-7. {$$ y = (x-1)^3(x+2)^2 $$}


D33-8. {$$ y = x^4 - 3x^2 - 9x +5 $$}

D33-9. {$$ y = x^4 $$}

D33-10. {$$ y = x^5 $$}

D33-11. {$$ y = x(x-1)(x-2) $$}

D33-12. {$$ y = {{8a} \over {x^2+4a^2}} $$}

D33-13. {$$ y = {1 \over {1+x^4}} $$}

D33-14. {$$ y = {1 \over {(1+x^2)^2}} $$}

D33-15. Show {$$ y = {{a^2x} \over {x^2+a^2}}$$} has three points of inflection lying on a straight line.

D33-16. Show for {$ y = x^n, n=1, 2, 3 \dots $} the origin is a minima or point of inflection depending upon whether n is even or odd.


  1. Elements of the Differential and Integral Calculus Wikisource
  2. Calculus Made Easy by Silvanus P. Thompson Project Gutenberg:
  3. Davis,Ellery Williams, William Charles Brenke, Earle Raymond Hedrick. The Calculus (Macmillan Company, 1922) Google Books
  4. Love, Clyde E., Earl David Rainville Differential and Integral Calculus (Macmillian, 1916) Google Books


  1. Paul's Online Notes: Calculus I'' while not a source of material here, sometimes helpful when stuck.

Category: Math Calculus

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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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December 23, 2018

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