The Mean Value Theorem

1. Which of the following functions satisfy the hypotheses of the Mean Value Theorem on the given intervals? Explain.     solution

(a)   \( f(x)=\dfrac{x^2-1}{x-1}, \:\: [0,2]\)       (b)   \( f(x)= |x|, \:\: [-1,1]\)      (c)   \(f(x)=\dfrac{1}{x-1}, \: \: [2,4]\)      (d)   \( f(x)= \ln{x}, \:\: [-1,2] \)      (e)   \( f(x)= \sqrt{x}, \:\: [-1,1]\)

2. The Mean Value Theorem guarantees the existence of a special number c in the interval \((0,4) \) for the function \(f(x)=\sqrt{x}\). Find the number c.      solution

3. The Mean Value Theorem guarantees the existence of a special number c in the interval \((0,16) \) for the function \(f(x)=\sqrt{x}\). Find the number c.    solution

4. Why is the function \( f(x)=3x^2+2x+5\) continuous on \([-1,1]\) and differentiable on \((-1,1)\)? Find all numbers c in \((-1,1)\) satisfying the conclusion of the Mean Value Theorem.     solution

5. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? If it does, find all numbers \(c \) that satisfy the conclusion of the Mean Value Theorem.   \(f(x)=\ln (2x), \quad [1/2,16]\)    solution

6. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? If it does, find all numbers \(c \) that satisfy the conclusion of the Mean Value Theorem.  \( \displaystyle{ f(x)=\frac{x}{x+9}, \quad[1,18]}\).    solution