Geometry of Functions VII: The First Derivative Test
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If you have been following our Geometry of Functions series then the First Derivative Test should almost be obvious. The First Derivative Test is a method for determining whether a function has a maximum or minimum in some interval. Here it is:
| The First Derivative Test |
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Let's assume that $e$ is a critical point of $f(x)$ and that $f(x)$ is continuous at $e$. Next, for some <Sign in to see all the formulas> we will look at the function's derivatives in the open interval <Sign in to see all the formulas>. If there is an interval <Sign in to see all the formulas> such that
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Let's jump into three examples that demonstrate the three scenarios. First, let's look at a maximum with the function <Sign in to see all the formulas>. We have found that the only critical point of <Sign in to see all the formulas> is $x=0$. We know that the derivative of <Sign in to see all the formulas> is <Sign in to see all the formulas>. Now when <Sign in to see all the formulas> then <Sign in to see all the formulas>. Also, when <Sign in to see all the formulas> then <Sign in to see all the formulas>. The First Derivative Test tells us that the critical point is at a maximum. See the picture below.
Next, let's look at a minimum with the function <Sign in to see all the formulas>. We have found that the only critical point of <Sign in to see all the formulas> is $x=0$. We know that the derivative of <Sign in to see all the formulas> is <Sign in to see all the formulas>. Now when <Sign in to see all the formulas> then <Sign in to see all the formulas>. Also, when <Sign in to see all the formulas> then <Sign in to see all the formulas>. The First Derivative Test tells us that the critical point is at a minimum. See the picture below.
Finally, let's look at the function <Sign in to see all the formulas>. We have found that the only critical point of <Sign in to see all the formulas> is $x=0$. We also know that the derivative of <Sign in to see all the formulas> is <Sign in to see all the formulas>. When <Sign in to see all the formulas> then <Sign in to see all the formulas> and when <Sign in to see all the formulas> then <Sign in to see all the formulas>. The First Derivative Test tells us that the critical point is neither a maximum nor a minimum. See the picture below.
