Theorems

Extreme Value Theorem

• Condition: $f(x)$ is continuous on interval $[a,b]$
• Conclusion: There must be an absolute maximum and an absolute minimum that must occur at one of the endpoints or at one of the relative maxima or minima
• How to apply
  1. Take critical values and endpoint x-values and make a table
  2. Plug each candidate into the function
  3. Determine the highest and lowest value

Mean Value Theorem

• Conditions
  • $f(x)$ is continuous on $[a,b]$
  • $f(x)$ is differentiable on $(a,b)$
• Conclusion
  • There must be at least one $c$ value on the interval where $f^{\prime }(c)$ is the average rate of change
• How to apply
  1. Find AROC on $[a,b]$
  2. Find $f^{\prime }(x)$ and set it equal to the AROC value
  3. Solve for $x$ (this is the $c$ value)

Rolle’s Theorem

• Conditions
  • $f(x)$ is continuous on $[a,b]$
  • $f(x)$ is differentiable on $(a,b)$
  • $f(a)=f(b)$
• Conclusion
  • There must be at least one $c$ value on the interval such that $f^{\prime }(c)=0$
• Just a special case of MVP