Q. 8 - Calculus

Question

Describe what the first-derivative test is for and how to use it. Sketch graphs and sign charts to illustrate your description.

Step-by-Step Solution

Answer

The first derivative test states that $f$ is continuous and differentiable at every point on $(a,b)$.

Step 1. Given Information.

Using the first derivative test and showing the graphs for the various descriptions.

Step 2. First derivative test.

Suppose $x=c$ is the location of a critical point of a function $f$, and let $(a,b)$ be an open interval around $c$ that is contained in the domain of $f$ and does not contain any other critical points of $f$.

If $f$ is continuous on $(a,b)$ and differentiate at every point of $(a,b)$ except possibly at $x=c$, then the following statements hold.

Step 3. Graphical Representation for the various ways.

(a) If $f\text{'}\left(x\right)$ is positive for $x?(a,c)$ and negative for $x?(c,b)$, then $f$ has a local maximum at $x=c$.

The first derivative $f\text{'}$ changes from positive to negative at $x=c$.

(b) If $f\text{'}\left(x\right)$ is negative for $x?(a,c)$ and positive for $x?(c,b)$, then $f$ has a local minimum at $x=c$.

The first derivative $f\text{'}$ changes from negative to positive at $x=c$.

(c) If $f\text{'}\left(x\right)$ is positive for both $x?(a,c)$ and $x?(c,b)$, then $f$ does not have a local extremum at $x=c$.

The first derivative $f\text{'}$is positive on both sides of $x=c$.

(d) If $f\text{'}\left(x\right)$ is negative for both $x?(a,c)$ and $x?(c,b)$, then $f$ does not have a local extremum at $x=c$.

The first derivative $f\text{'}$ is negative on both sides of $x=c$.