Shape of a Graph            Practice Curve Sketching

  1. Consider \(f(x) = x^3 + 3x^2 - 9x + 10\). Answer the following using calculus.  solution

    2. (a)  Find the intervals on which \( f \) is increasing or decreasing.

    (b)  Find the \(x\)-values where \(f\) attains its local maximum and minimum values.

  2. Find the local maximum and local minimum values of \(f(x)=x^2e^x \). solution


  3. Consider the function \( f(x)=5x^{\frac{1}{3}} - x^{\frac{5}{3}}\). Find the intervals where \(f\) is increasing or decreasing. Also find the local maxima and minima.   solution


  4. Let \(f(x) = x^3 + 3x^2 - 24x + 5\). Answer the following.     solution

    5. a. Find all the critical numbers of \( f \).

    b. Find the intervals on which \( f \) is increasing or decreasing.

    c. Find the local maximum and minimum values of \( f \).

    d. Find the intervals on which \( f \) is concave up or down.

    e. Find the inflection points of \( f \).

     

  5. Consider \( f(x)=2x^3-3x^2\). Answer the following using calculus.      solution

    6. a. Find the intervals on which \( f \) is concave up or down.

    b. Find the inflection points of \( f \).


  6. Consider the function \( f(x)=x + x^2 - x^3\). Answer the following using calculus.  solution

    7. 1. Find the intervals on which \(f\) is concave up or concave down.

    2. Find the \(x\)-coordinate(s) of inflection point(s) of \(f\).


  7. (Two problems) Use the graph of \(y=f'(x)\) below to answer the following questions about the function \(f\).  Solution of graph A   Solution of graph B

    8. (A) graph2

    (B) new graph

    (a) On what interval(s) is the graph of \(f\) increasing or decreasing? Justify your answer.

    (b) Find the \(x\)-value(s) at which f has a local maximum or minimum, and state which is which.

    (c) On which interval(s) is the graph of \( f \) concave up or concave down?

    (d) State the \(x\)-coordinates(s) of inflection point(s) of \( f \), if any.


  8. The graph of the derivative function \( y=f'(x) \) is given below. Answer the following questions about \( y=f(x)\).  solution

    9. graph

    a. Find all the critical numbers of \( f \)

    b. On what intervals is the graph of \( f \) increasing or decreasing?

    c. Find the \(x\)-value(s) at which \( f \) has a local maximum or minimum, and identify which is which.

    d. On which interval(s) is the graph of \( f \) concave up or concave down?

    e. State the \(x\)-coordinates(s) of inflection point(s) of \( f \), if any.

     

  9. The graph of the second derivative \( \, f''\) of a function \(f\) shown. Find the \(x\)-coordinates of the inflection points of \(f\).    solution

    10. fprme