Exercises - The Mean Value Theorem

  1. Given $f\,(x) = \sqrt{25-x^2}$, show that the Mean Value Theorem applies to this function over the interval $[-3,5]$, and then do the following:

    1. Sketch the graph of this function and label the ordered pairs that determine the chord relevant to the application of the Mean Value Theorem
    2. Find the value $c$ guaranteed to exist by this theorem
    3. Place $(c,f\,(c))$ on the sketch and show that the tangent line is parallel to the chord

  2. Determine if the Mean Value Theorem applies or does not apply to the given function on the given interval. When it applies, state clearly its conclusion, and then find the value(s) guaranteed to exist by the theorem. When it does not apply, state why.

    1. $\displaystyle{f\,(x) = x^4 - 4x; \quad [2,6]}$  

    2. $\displaystyle{f\,(x) = x^3 - x^2; \quad [-1,2]}$

    3. $\displaystyle{f\,(x) = \frac{1}{x^3}; \quad [-1,1]}$

    4. $\displaystyle{f\,(x) = |x+2|; \quad [-3,0]}$

    5. $\displaystyle{f\,(x) = x^3-3x^2+2x; \quad [0,2]}$

    6. $\displaystyle{f\,(x) = 8 - x^3; \quad [-2,1]}$

    7. $\displaystyle{f\,(x) = 3x^{5/3} - 5x; \quad [-1,1]}$

    8. $\displaystyle{f\,(x) = \frac{1}{x-1}; \quad [-2,1]}$

    9. $\displaystyle{f\,(x) = x^2 - 5; \quad [-3,0]}$

    10. $\displaystyle{f\,(x) = \sqrt{4-x^2}; \quad [0,2]}$

    11. $\displaystyle{f\,(x) = 5x^{2/3} - 5; \quad [-1,1]}$

    12. $\displaystyle{f\,(x) = x^{1/3} - 1; \quad [-1,1]}$

    13. $\displaystyle{f\,(x) = x^3 - 1; \quad [-2,2]}$

    14. $\displaystyle{f\,(x) = x^3 + 3x^2; \quad [-2,0]}$