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Home / Calculus I / Applications of Derivatives / The Shape of a Graph, Part I
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Section 4.5 : The Shape of a Graph, Part I

For problems 1 – 4 the graph of a function is given. Determine the intervals on which the function increases and decreases.

  1. This graph has no y scale. It starts at x=-6 in the second quadrant that is the highest point on the graph. The graph has a valley at x=-4 in the second quadrant, followed by a peak at x=-2 in the second quadrant, then a valley at x=5 in the 4th quadrant and ending at a point in fourth quadrant at x=7.
  2. This graph has no y scale. It starts at x=-4 in the second quadrant that is the followed by a peak at x=-3 in the second quadrant. The graph then has a valley at x=1 in the first quadrant, followed by a peak at x=3 in the first quadrant, then a valley at x=6 in the 4th quadrant and ending at a point in slightly above the x-axis in the first quadrant at x=7.
  3. This graph has no y scale. It starts at x=-7 in the second quadrant that is the followed by a peak at x=-6 in the second quadrant. The graph then goes through a point at x=-2 perfectly flat and then has a valley at x=1 in the fourth quadrant and ending at a point in the first quadrant at x=2.
  4. This graph has no y scale. It starts at approximately x=-2.5 on the x-axis and is the followed by a peak at x=-2 in the second quadrant. The graph then goes through the origin perfectly flat and then has a valley at x=2 in the fourth quadrant. It then goes through a point at x=4 perfectly flat and ends at a point in the first quadrant at x=5 that is the highest point on the graph.

For problems 5 – 7 the graph of the derivative of a function is given. From this graph determine the intervals in which the function increases and decreases.

  1. This graph has no y scale. It starts in the fourth quadrant at x=-9 and goes through the x-axis at x=-8 reaching a peak at approximately x=5. It then goes through the axis at x=-1 and reaches a valley at approximately x=1. Finally it goes back through the x-axis at x=3 and ends at a point in the first quadrant at x=4.
  2. This graph has no y scale. It starts in the second quadrant at x=-6 and goes down and just touches the x-axis at x=-5 and then is followed by a peak at approximately x=-3.8. It then goes through the axis at x=-3 and reaches a valley at approximately x=-1. Finally, it goes through the origin and ends at a point in the first quadrant at approximately x=0.5.
  3. This graph has no y scale. It starts in the fourth quadrant at approximately x=-2.5 and goes up and just touches the x-axis at x=-2 and then is followed by a valley at approximately x=-0.8. It then goes back up to just touch the x-axis at x=1 followed by another valley at approximately x=2.2. It goes back up to just touch the x-axis again at x=3 and then ends at a point in the fourth quadrant at approximately x=3.5.

For problems 8 – 10 The known information about the derivative of a function is given. From this information answer each of the following questions.

  1. Identify the critical points of the function.
  2. Determine the intervals on which the function increases and decreases.
  3. Classify the critical points as relative maximums, relative minimums or neither.

  1. \[\begin{array}{c}f'\left( 1 \right) = 0\hspace{0.5in}f'\left( 3 \right) = 0\hspace{0.5in}f'\left( 8 \right) = 0\\ f'\left( x \right) < 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( { - \infty ,1} \right),\,\,\,\left( {3,8} \right)\hspace{0.25in}f'\left( x \right) > 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( {1,3} \right),\,\,\,\left( {8,\infty } \right)\end{array}\]
  2. \[\begin{array}{c}g'\left( { - 2} \right) = 0\hspace{0.5in}\,g'\left( 0 \right) = 0\hspace{0.5in}g'\left( 3 \right) = 0\hspace{0.5in}\,g'\left( 6 \right) = 0\\ g'\left( x \right) < 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( {0,3} \right),\,\,\,\left( {6,\infty } \right)\hspace{0.5in}g'\left( x \right) > 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( { - \infty , - 2} \right),\,\,\,\left( { - 2,0} \right),\,\,\,\left( {3,6} \right)\end{array}\]
  3. \[\begin{array}{c}h'\left( { - 1} \right) = 0\hspace{0.5in}\,h'\left( 2 \right) = 0\hspace{0.5in}h'\left( 5 \right) = 0\\ h'\left( x \right) < 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( { - \infty , - 1} \right),\,\,\,\left( { - 1,2} \right)\hspace{0.5in}h'\left( x \right) > 0\,\,\,\,\,{\mbox{on}}\,\,\,\,\,\left( {2,5} \right),\,\,\,\left( {5,\infty } \right)\end{array}\]

For problems 11 – 28 answer each of the following.

  1. Identify the critical points of the function.
  2. Determine the intervals on which the function increases and decreases.
  3. Classify the critical points as relative maximums, relative minimums or neither.

  1. \(f\left( t \right) = {t^3} - 15{t^2} + 63t + 3\)
  2. \(g\left( x \right) = 20 + 8{x^2} + 4{x^3} - {x^4}\)
  3. \(Q\left( w \right) = 8{w^3} - 18{w^2} - 24w - 10\)
  4. \(\displaystyle f\left( x \right) = {x^5} + \frac{5}{4}{x^4} - 20{x^3} - 7\)
  5. \(P\left( x \right) = 5 - 4x - 9{x^2} - 3{x^3}\)
  6. \(R\left( z \right) = {z^5} + {z^4} - 6{z^3} + 5\)
  7. \(h\left( z \right) = 1 - 12{z^2} - 9{z^3} - 2{z^4}\)
  8. \(Q\left( t \right) = 7 - t + \sin \left( {4t} \right)\)on \(\displaystyle \left[ - \frac{3}{2},\frac{3}{2} \right]\)
  9. \(\displaystyle f\left( z \right) = 6z - 20\cos \left( \frac{z}{2} \right)\) on \(\left[ {0,22} \right]\)
  10. \(\displaystyle g\left( x \right) = 24\cos \left( \frac{x}{3} \right) + 8x + 2\) on \(\left[ { - 30,25} \right]\)
  11. \(h\left( w \right) = 9w - 5\sin \left( {2w} \right)\) on \(\left[ { - 5,0} \right]\)
  12. \(h\left( x \right) = \sqrt[5]{x}\,\,\left( {x + 7} \right)\)
  13. \(W\left( z \right) = \left( {10 - {w^2}} \right){\left( {w + 2} \right)^{\frac{2}{3}}}\)
  14. \(f\left( t \right) = \left( {{t^2} - 8} \right)\,\,\,\,\sqrt[3]{{{t^2} - 4}}\)
  15. \(f\left( x \right) = {{\bf{e}}^{\frac{1}{3}{x^{\,3}} - {x^{\,2}} - 3x}}\)
  16. \(h\left( z \right) = \left( {{z^2} - 8} \right){{\bf{e}}^{3\, - z}}\)
  17. \(A\left( t \right) = \ln \left( {{t^2} + 5t + 8} \right)\)
  18. \(g\left( x \right) = x - 3 + \ln \left( {1 + x + {x^2}} \right)\)
  19. Answer each of the following questions.
    1. What is the minimum degree of a polynomial that has exactly one relative extrema?
    2. What is the minimum degree of a polynomial that has exactly two relative extrema?
    3. What is the minimum degree of a polynomial that has exactly three relative extrema?
    4. What is the minimum degree of a polynomial that has exactly \(n\) relative extrema?
  20. For some function, \(f\left( x \right)\), it is known that there is a relative minimum at \(x = - 4\). Answer each of the following questions about this function.
    1. What is the simplest form that the derivative of this function?
      Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative.
    2. Using your answer from (a) determine the most general form that the function itself can take.
    3. Given that \(f\left( { - 4} \right) = 6\) find a function that will have a relative minimum at \(x = - 4\).
      Note : There are many possible answers here so just give one of them.
  21. For some function, \(f\left( x \right)\), it is known that there is a relative maximum at \(x = - 1\). Answer each of the following questions about this function.
    1. What is the simplest form that the derivative of this function?
      Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative.
    2. Using your answer from (a) determine the most general form that the function itself can take.
    3. Given that \(f\left( { - 1} \right) = 3\) find a function that will have a relative maximum at \(x = - 1\).
      Note : There are many possible answers here so just give one of them.
  22. For some function, \(f\left( x \right)\), it is known that there is a critical point at \(x = 3\) that is neither a relative minimum or a relative maximum. Answer each of the following questions about this function.
    1. What is the simplest form that the derivative of this function?
      Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative.
    2. Using your answer from (a) determine the most general form that the function itself can take.
    3. Given that \(f\left( 3 \right) = 2\) find a function that will have a critical point at \(x = 3\) that is neither a relative minimum or a relative maximum.
      Note : There are many possible answers here so just give one of them.
  23. For some function, \(f\left( x \right)\), it is known that there is a relative maximum at \(x = 1\) and a relative minimum at \(x = 4\). Answer each of the following questions about this function.
    1. What is the simplest form that the derivative of this function?
      Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative.
    2. Using your answer from (a) determine the most general form that the function itself can take.
    3. Given that \(f\left( 1 \right) = 6\) and \(f\left( 4 \right) = - 2\) find a function that will have a relative maximum at \(x = 1\) and a relative minimum at \(x = 4\).
      Note : There are many possible answers here so just give one of them.
  24. Given that \(f\left( x \right)\) and \(g\left( x \right)\) are increasing functions will \(h\left( x \right) = f\left( x \right) - g\left( x \right)\) always be an increasing function? If so, prove that \(h\left( x \right)\) will be an increasing function. If not, find increasing functions, \(f\left( x \right)\) and \(g\left( x \right)\), so that \(h\left( x \right)\) will be a decreasing function and find a different set of increasing functions so that \(h\left( x \right)\) will be an increasing function.
  25. Given that \(f\left( x \right)\) is an increasing function. There are several possible conditions that we can impose on \(g\left( x \right)\) so that \(h\left( x \right) = f\left( x \right) - g\left( x \right)\) will be an increasing function. Determine as many of these possible conditions as you can.
  26. For a function \(f\left( x \right)\) determine a set of conditions on \(f\left( x \right)\), different from those given in #15 in the practice problems, for which \(h\left( x \right) = {\left[ {f\left( x \right)} \right]^2}\) will be an increasing function.
  27. For a function \(f\left( x \right)\) determine a single condition on \(f\left( x \right)\) for which \(h\left( x \right) = {\left[ {f\left( x \right)} \right]^3}\) will be an increasing function.
  28. Given that \(f\left( x \right)\) and \(g\left( x \right)\) are positive functions. Determine a set of conditions on them for which \(h\left( x \right) = f\left( x \right)g\left( x \right)\) will be an increasing function. Note that there are several possible sets of conditions here but try to determine the “simplest” set of conditions.
  29. Repeat #38 for \(\displaystyle h\left( x \right) = \frac{{f\left( x \right)}}{{g\left( x \right)}}\).
  30. Given that \(f\left( x \right)\) and \(g\left( x \right)\) are increasing functions prove that \(h\left( x \right) = f\left( {g\left( x \right)} \right)\) will also be an increasing function.
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