Q. 1 C - Calculus

Question

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: If $f\text{'}\text{'}\left(2\right)=0$, then $x=2$ is an inflection of point $f$.

(b) True or False: If $f\text{'}\text{'}$ is concave up on an interval $I$, then it is positive on $I$.

(c) True or False: If $f$ is concave up on an interval $I$, then $f\text{'}\text{'}$ is positive on $I$.

(d) True or False: If $f\text{'}\text{'}\left(2\right)$ does not exist and $x=2$ is in the domain  of $f$, then $x=2$ is critical point of the  function $f\text{'}.$

(e) True or False: : If $f$ has an inflection point at $x=3$ and $f$ is differentiable at $x=3$, then the derivative $f\text{'}$ has a local minimum or maximum at $x=3$

(f) True or False: If $f\text{'}\left(1\right)=0 and f\text{'}\text{'}\left(1\right)=-2, then f$ has a local minimum at $x=1$

(g) True or False: The second-derivative test involves checking the sign of the second derivative on each side of every critical point.

(h) True or False: The second-derivative test always produces exactly the same information as the first-derivative test.

Step-by-Step Solution

Answer

Part (a): False

Part (b): False 

Part (c): False 

Part (d): True

Part (e): True

Part (f): False 

Part (g): False 

Part (h): False 

Part (a) Step 1. Explanation

Inflection points of a function are the points in the domain of $f$ at which its concavity changes. Since the sign of $f\text{'}$ measures the concavity of $f$, you can find the inflection points by looking for the places where $f\text{'}$ changes sign.

If $x=c$ is an inflection point of $f and f\text{'}\left(c\right)=0,$then the graph of would be near $x=c$, depending on how $f$ changes concavity and whether $f$ is an increasing or decreasing.

Thus the given statement is false,

Part (b) Step 1. Explanation

Suppose $f and f\text{'}$ are differentiable on an interval $I,$then

If $f\text{'}$ is positive on $I, then f$ is concave up on $I$.

Again,

If $f\text{'}$ is negative on $I$, then $f$ is concave down on I.

Thus, the given statement is false.

Part (c) Step 1. Explanation

Suppose both $f$ and $f\text{'}$ are differentiable on an interval I, then

If $f\text{'}$ is positive on I, then $f$ is concave up on I.

Again,

If $f\text{'}$ is negative on I, then $f$ is concave down on I.

Thus the given statement is false.

Part (d) Step 1. Explanation

Suppose $x=c$ is the location of a critical point of a function $f$ with $f\text{'}\left(c\right)=0$, and suppose both $f$ and $f\text{'}$ are differentiable and $f\text{'}$ is continuous on an interval around $x=c$ then,

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

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

And

If $f\text{'}\left(c\right)=0$, then this test says that nothing about whether or not has extremum at $x=c$

Thus the given statement is true.

Part (e) Step 1. Explanation

$f\text{'}\left(c\right)=0$Suppose $x=c$ is the location of a critical point of a function $f$ with $f\text{'}\left(c\right)=0$, and suppose both $f and f\text{'}$ are differentiable and $f\text{'}$ is continuous on an interval around $x=c$, then

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

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

And 

If $f\text{'}\left(c\right)=0,$then this test says that nothing about whether or not $f$ has extremum at $x=c$

Thus, the given statement is true.

Part (f) Step 1. Explanation

Suppose $x=c$ is the location of a critical point of a function $f$ with $f\text{'}\left(c\right)=0$, and suppose both $f and f\text{'}$are differentiable and $f\text{'}$ is continuous on an interval around $x=c$, then

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

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

And

If $f\text{'}\left(c\right)=0,$ then this test says that nothing about whether or not $f$ has extremum at $x=c$.

Thus the given statement is false.

Part (g) Step 1. Explanation

Suppose $x=c$ is the location of a critical point of a function $f$ with $f\text{'}\left(c\right)=0$, and suppose both  $f$ and $f\text{'}$ are differentiable and $f\text{'}$is continuous on an interval around $x=c$, then

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

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

And

If $f\text{'}\left(c\right)=0$, then this test says that nothing about whether or not $f$ has extremum at $x=c$.

Thus the given statement is false.

Part (h) Step 1. Explanation

Suppose $x=c$is the location of a critical point of a function $f$ with $f\text{'}\left(c\right)=0$, and suppose both $f$ and $f\text{'}$ are differentiable and $f\text{'}$ is continuous on an interval around $x=c$, then

If $f\text{'}$ is positive, then $f$ has a local minimum at $x=c$.

If $f\text{'}$ is negative, then $f$ has a local maximum at $x=c$.

And

If $f\text{'}\left(c\right)=0$, then this test says that nothing about whether or not $f$ has extremum at $x=c$.

Thus the given statement is false.