Prove Theorem 2 in the lesson 'Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions'.

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Lesson Parent: 
Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions
Problem: 

Assume that $f(x)$ is continuous on a closed interval $I$ and differentiable on the interior of $I$. Prove that

  • If $f\,'(x)\gt 0$ for all $x$ in the interior of $I$, then $f(x)$ increases on $I$.
  • If $f\,'(x)\lt 0$ for all $x$ in the interior of $I$, then $f(x)$ decreases on $I$.
  • If $f\,'(x)=0$ for all $x$ in the interior of $I$, then $f(x)$ is constant on $I$.
Answer: 

SAMPLE SOLUTION

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Solution: 

We assume that $f(x)$ is continuous on a closed interval $I$ and differentiable on the interior of $I$. Let's denote our closed interval by <Sign in to see all the formulas> and arbitrarily pick two numbers $x_1$ and $x_2$ in the interior of $I$ such that <Sign in to see all the formulas>. So now we know that <Sign in to see all the formulas>. Because $f(x)$ is differentiable on $(a,b)$ we know that it is continuous on $(a,b)$. Therefore, it is continuous at $x_1$ and $x_2$. Because it is differentiable on <Sign in to see all the formulas> and continuous on <Sign in to see all the formulas>, we have satisfied the requirements for using the Mean-Value Theorem. Therefore, we know that there is some number $e$ in <Sign in to see all the formulas> such that

<Sign in to see all the formulas>

Now we can begin the three proofs. Let's do 1. Since we assume that <Sign in to see all the formulas> then

<Sign in to see all the formulas>


Since <Sign in to see all the formulas> then <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. Therefore, $f(x)$ satisfies the definition of an increasing function.

Now for proof 2. Since we assume that <Sign in to see all the formulas> then

<Sign in to see all the formulas>


Since <Sign in to see all the formulas> then <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. Therefore, $f(x)$ satisfies the definition of a decreasing function.

Finally, we have proof 3. Since we assume that <Sign in to see all the formulas> then

<Sign in to see all the formulas>


Therefore, <Sign in to see all the formulas>.

We have shown proofs 1-3 are true for an arbitrarily picked value of $x_1$ and $x_2$. Since the values we have picked are arbitrary the proofs are therefore true for all values of $x_1$ and $x_2$ in $I$.