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AP Calculus BC : Inflection Points

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Example Question #11 :Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=6x - 8x^{2} - 3x^{3} - 5\end{align*}$

Possible Answers:

-4.444

-1.778

$\text{There are no points of inflection.}$

-0.889

Correct answer:

-0.889

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&6 - 9x^{2} - 16x\\&\text{And our second is:}\\&- 18x - 16\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.88889\\&\text{We find one real unique root at the point }x=-0.889\\&\text{Now to determine if a point is one of inflection, check values on either side of it. Say }\pm0.005/\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #11 :Derivatives

$\begin{align*}&\text{Determine what, if any exist, the points of inflection are for the function}\\&y=6x^{5} - 7x^{2} - 8x^{3} - 5x^{4} - 10x + 1\end{align*}$

Possible Answers:

5.051

1.010

3.031

$\text{There are no points of inflection.}$

Correct answer:

1.010

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&30x^{4} - 24x^{2} - 20x^{3} - 14x - 10\\&\text{And our second is:}\\&120x^{3} - 60x^{2} - 48x - 14\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=1.0103,-0.25513+0.22449i,-0.25513-0.22449i\\&\text{We find one real unique root at the point }x=1.010\\&\text{Now to determine if a point is one of inflection, check values on either side of it. Say }\pm0.005/\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #11 :Second Derivatives

$\begin{align*}&\text{Determine what, if any exist, the points of inflection are for the function}\\&y=8x - 4x^{2} + 3x^{3} + 4x^{4} - 3\end{align*}$

Possible Answers:

-1.273,0.785

$\text{There are no points of inflection.}$

-0.637,0.262

-3.820,1.570

Correct answer:

-0.637,0.262

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&9x^{2} - 8x + 16x^{3} + 8\\&\text{And our second is:}\\&18x + 48x^{2} - 8\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.637,0.262\\&\text{We find real unique roots at the points }x=-0.637,0.262\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #11 :Inflection Points

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=1 - 10x^{3} - 7x^{4} - 4x^{2}\end{align*}$

Possible Answers:

-3.221,-1.242

$\text{There are no points of inflection.}$

-2.148,-0.710

-0.537,-0.177

Correct answer:

-0.537,-0.177

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&- 8x - 30x^{2} - 28x^{3}\\&\text{And our second is:}\\&- 60x - 84x^{2} - 8\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.537,-0.177\\&\text{We find real unique roots at the points }x=-0.537,-0.177\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #11 :Derivatives

$\begin{align*}&\text{Determine what, if any exist, the points of inflection are for the function}\\&y=9x^{2} - x - 4x^{3} - 8x^{4} - 5\end{align*}$

Possible Answers:

$\text{There are no points of inflection.}$

-0.576,0.326

-1.151,0.651

-3.454,1.628

Correct answer:

-0.576,0.326

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&18x - 12x^{2} - 32x^{3} - 1\\&\text{And our second is:}\\&18 - 96x^{2} - 24x\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.576,0.326\\&\text{We find real unique roots at the points }x=-0.576,0.326\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #11 :Second Derivatives

$\begin{align*}&\text{Determine what, if any exist, the points of inflection are for the function}\\&y=4x^{3} - 3x - 10x^{4} + x^{5} + 7\end{align*}$

Possible Answers:

0.000,1.036,28.964

0.000,0.414,23.171

$\text{There are no points of inflection.}$

0.000,0.207,5.793

Correct answer:

0.000,0.207,5.793

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&12x^{2} - 40x^{3} + 5x^{4} - 3\\&\text{And our second is:}\\&24x - 120x^{2} + 20x^{3}\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.000,0.207,5.793\\&\text{We find real unique roots at the points }x=0.000,0.207,5.793\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #21 :Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=x^{2} - 6x + 2x^{3} - 2x^{4} + 6\end{align*}$

Possible Answers:

-0.132,0.632

-0.264,2.528

$\text{There are no points of inflection.}$

-0.923,3.159

Correct answer:

-0.132,0.632

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&2x + 6x^{2} - 8x^{3} - 6\\&\text{And our second is:}\\&12x - 24x^{2} + 2\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.132,0.632\\&\text{We find real unique roots at the points }x=-0.132,0.632\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #21 :Second Derivatives

$\begin{align*}&\text{Determine what, if any exist, the points of inflection are for the function}\\&y=5x^{3} - 10x^{2} - 10x - 5\end{align*}$

Possible Answers:

0.667

2.000

4.667

$\text{There are no points of inflection.}$

Correct answer:

0.667

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&15x^{2} - 20x - 10\\&\text{And our second is:}\\&30x - 20\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.667\\&\text{We find one real unique root at the point }x=0.667\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #21 :Second Derivatives

$\begin{align*}&\text{Find any points of inflection for the function:}\\&7x - 2x^{2} - 2x^{3} + 6\end{align*}$

Possible Answers:

-0.333

-1.333

$\text{There are no points of inflection.}$

-2.000

Correct answer:

-0.333

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&7 - 6x^{2} - 4x\\&\text{And our second is:}\\&- 12x - 4\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.333\\&\text{We find one real unique root at the point }x=-0.333\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #21 :Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=8x - 9x^{2} + 3x^{3} + 8x^{4} + 8x^{5} + 8\end{align*}$

Possible Answers:

1.188

2.079

$\text{There are no points of inflection.}$

0.297

Correct answer:

0.297

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&9x^{2} - 18x + 32x^{3} + 40x^{4} + 8\\&\text{And our second is:}\\&18x + 96x^{2} + 160x^{3} - 18\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.297,-0.448+0.422i,-0.448-0.422i\\&\text{We find one real unique root at the point }x=0.297\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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AP Calculus BC : Inflection Points

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #11 :Second Derivatives

Example Question #11 :Derivatives

Example Question #11 :Second Derivatives

Example Question #11 :Inflection Points

Example Question #11 :Derivatives

Example Question #11 :Second Derivatives

Example Question #21 :Derivatives

Example Question #21 :Second Derivatives

Example Question #21 :Second Derivatives

Example Question #21 :Derivatives

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