Concavity of Functions Over Their Domains - AP Calculus AB
Card 1 of 30
Find the concavity of $f(x) = \text{ln}(x)$ at $x = 2$.
Find the concavity of $f(x) = \text{ln}(x)$ at $x = 2$.
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$f''(x) = -\frac{1}{x^2}$, $f''(2) < 0$, concave down. Natural logarithm has negative second derivative.
$f''(x) = -\frac{1}{x^2}$, $f''(2) < 0$, concave down. Natural logarithm has negative second derivative.
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Determine concavity of $f(x) = x^3 + 6x^2 + 9x$ at $x = -2$.
Determine concavity of $f(x) = x^3 + 6x^2 + 9x$ at $x = -2$.
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$f''(x) = 6x + 12$, $f''(-2) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
$f''(x) = 6x + 12$, $f''(-2) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
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What is the concavity of $f(x) = x^2$ over its domain?
What is the concavity of $f(x) = x^2$ over its domain?
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$f''(x) = 2 > 0$, so $f(x)$ is concave up everywhere. The second derivative is constant and positive.
$f''(x) = 2 > 0$, so $f(x)$ is concave up everywhere. The second derivative is constant and positive.
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At what point does concavity change for $f(x) = x^3$?
At what point does concavity change for $f(x) = x^3$?
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Concavity changes at $x = 0$, where $f''(x) = 0$. Inflection occurs where $f''(x) = 0$ and changes sign.
Concavity changes at $x = 0$, where $f''(x) = 0$. Inflection occurs where $f''(x) = 0$ and changes sign.
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What is the concavity of $f(x) = x^3 + 3x^2$ at $x = -1$?
What is the concavity of $f(x) = x^3 + 3x^2$ at $x = -1$?
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$f''(x) = 6x + 6$, $f''(-1) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
$f''(x) = 6x + 6$, $f''(-1) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
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Identify concavity of $f(x) = \text{arctan}(x)$ at $x = 0$.
Identify concavity of $f(x) = \text{arctan}(x)$ at $x = 0$.
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$f''(x) = -\frac{2x}{(1+x^2)^2}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.
$f''(x) = -\frac{2x}{(1+x^2)^2}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.
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Identify an inflection point for $f(x) = x^3 - 3x^2 + x$.
Identify an inflection point for $f(x) = x^3 - 3x^2 + x$.
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Inflection at $x = 1$ where $f''(x)$ changes sign. Second derivative changes sign at this point.
Inflection at $x = 1$ where $f''(x)$ changes sign. Second derivative changes sign at this point.
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Determine concavity of $f(x) = e^x$ over its domain.
Determine concavity of $f(x) = e^x$ over its domain.
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$f''(x) = e^x > 0$, so $f(x)$ is concave up everywhere. The exponential function has constant positive concavity.
$f''(x) = e^x > 0$, so $f(x)$ is concave up everywhere. The exponential function has constant positive concavity.
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Determine concavity of $f(x) = x^4 - 4x^2$ at $x = 1$.
Determine concavity of $f(x) = x^4 - 4x^2$ at $x = 1$.
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$f''(x) = 12x^2 - 8$, $f''(1) = 4 > 0$, concave up. Positive second derivative at the specified point.
$f''(x) = 12x^2 - 8$, $f''(1) = 4 > 0$, concave up. Positive second derivative at the specified point.
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What is the concavity of $f(x) = \text{ln}(x)$ over $x > 0$?
What is the concavity of $f(x) = \text{ln}(x)$ over $x > 0$?
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$f''(x) = -\frac{1}{x^2} < 0$, concave down for $x > 0$. Natural log is concave down on its domain.
$f''(x) = -\frac{1}{x^2} < 0$, concave down for $x > 0$. Natural log is concave down on its domain.
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Determine concavity of $f(x) = x^3 - x$ at $x = 1$.
Determine concavity of $f(x) = x^3 - x$ at $x = 1$.
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$f''(x) = 6x$, $f''(1) = 6 > 0$, concave up. Positive second derivative indicates upward concavity.
$f''(x) = 6x$, $f''(1) = 6 > 0$, concave up. Positive second derivative indicates upward concavity.
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Identify the concavity for $f(x) = \text{cos}(x)$ at $x = \frac{\text{π}}{2}$.
Identify the concavity for $f(x) = \text{cos}(x)$ at $x = \frac{\text{π}}{2}$.
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$f''(x) = -\text{cos}(x)$, $f''(\frac{\text{π}}{2}) = 0$, undetermined. When $f''(x) = 0$, concavity cannot be determined.
$f''(x) = -\text{cos}(x)$, $f''(\frac{\text{π}}{2}) = 0$, undetermined. When $f''(x) = 0$, concavity cannot be determined.
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What is the concavity of $f(x) = 4x^4 - x^2$ at $x = 0$?
What is the concavity of $f(x) = 4x^4 - x^2$ at $x = 0$?
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$f''(x) = 48x^2 - 2$, $f''(0) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.
$f''(x) = 48x^2 - 2$, $f''(0) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.
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What is the concavity of $f(x) = -x^2$ over its domain?
What is the concavity of $f(x) = -x^2$ over its domain?
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$f''(x) = -2 < 0$, so $f(x)$ is concave down everywhere. Constant negative second derivative everywhere.
$f''(x) = -2 < 0$, so $f(x)$ is concave down everywhere. Constant negative second derivative everywhere.
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Identify concavity of $f(x) = x^2 + 4$ at $x = 0$.
Identify concavity of $f(x) = x^2 + 4$ at $x = 0$.
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$f''(x) = 2 > 0$, concave up. Constant positive second derivative for all quadratics.
$f''(x) = 2 > 0$, concave up. Constant positive second derivative for all quadratics.
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What is the concavity of $f(x) = x^5$ at $x = 0$?
What is the concavity of $f(x) = x^5$ at $x = 0$?
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$f''(x) = 20x^3$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
$f''(x) = 20x^3$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
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Identify the concavity for $f(x) = \frac{1}{x}$ at $x = -1$.
Identify the concavity for $f(x) = \frac{1}{x}$ at $x = -1$.
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$f''(x) = \frac{2}{x^3}$, $f''(-1) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.
$f''(x) = \frac{2}{x^3}$, $f''(-1) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.
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How does the sign of $f''(x)$ affect the graph of $f(x)$?
How does the sign of $f''(x)$ affect the graph of $f(x)$?
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If $f''(x) > 0$, $f(x)$ is concave up; if $f''(x) < 0$, $f(x)$ is concave down. Sign of the second derivative directly determines concavity.
If $f''(x) > 0$, $f(x)$ is concave up; if $f''(x) < 0$, $f(x)$ is concave down. Sign of the second derivative directly determines concavity.
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Identify concavity of $f(x) = \text{tan}(x)$ at $x = 0$.
Identify concavity of $f(x) = \text{tan}(x)$ at $x = 0$.
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$f''(x) = 2\text{tan}(x)\text{sec}^2(x)$, $f''(0) = 0$. Undetermined. Zero second derivative requires further analysis.
$f''(x) = 2\text{tan}(x)\text{sec}^2(x)$, $f''(0) = 0$. Undetermined. Zero second derivative requires further analysis.
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Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$.
Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$.
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$f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.
$f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.
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Determine concavity of $f(x) = e^{-x}$ over its domain.
Determine concavity of $f(x) = e^{-x}$ over its domain.
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$f''(x) = e^{-x} > 0$, so $f(x)$ is concave up everywhere. Exponential decay function maintains positive concavity.
$f''(x) = e^{-x} > 0$, so $f(x)$ is concave up everywhere. Exponential decay function maintains positive concavity.
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Determine concavity of $f(x) = x^2 - 4x$ at $x = 3$.
Determine concavity of $f(x) = x^2 - 4x$ at $x = 3$.
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$f''(x) = 2 > 0$, concave up. Constant positive second derivative for quadratic functions.
$f''(x) = 2 > 0$, concave up. Constant positive second derivative for quadratic functions.
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Identify concavity of $f(x) = \text{sin}(x)$ at $x = 0$.
Identify concavity of $f(x) = \text{sin}(x)$ at $x = 0$.
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$f''(x) = -\text{sin}(x)$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
$f''(x) = -\text{sin}(x)$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.
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Identify concavity of $f(x) = x^3 - 3x$ at $x = -1$.
Identify concavity of $f(x) = x^3 - 3x$ at $x = -1$.
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$f''(x) = 6x$, $f''(-1) = -6 < 0$, concave down. Negative second derivative indicates downward concavity.
$f''(x) = 6x$, $f''(-1) = -6 < 0$, concave down. Negative second derivative indicates downward concavity.
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Identify concavity of $f(x) = \frac{1}{x^2}$ at $x = 1$.
Identify concavity of $f(x) = \frac{1}{x^2}$ at $x = 1$.
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$f''(x) = \frac{6}{x^4}$, $f''(1) = 6 > 0$, concave up. Positive second derivative confirms upward concavity.
$f''(x) = \frac{6}{x^4}$, $f''(1) = 6 > 0$, concave up. Positive second derivative confirms upward concavity.
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What is the concavity of $f(x) = x^4$ at $x = 1$?
What is the concavity of $f(x) = x^4$ at $x = 1$?
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$f''(x) = 12x^2$, $f''(1) = 12 > 0$, concave up. Positive second derivative indicates upward concavity.
$f''(x) = 12x^2$, $f''(1) = 12 > 0$, concave up. Positive second derivative indicates upward concavity.
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Determine concavity of $f(x) = \frac{1}{x}$ at $x = 1$.
Determine concavity of $f(x) = \frac{1}{x}$ at $x = 1$.
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$f''(x) = \frac{2}{x^3}$, $f''(1) = 2 > 0$, concave up. Positive second derivative at a point indicates concave up.
$f''(x) = \frac{2}{x^3}$, $f''(1) = 2 > 0$, concave up. Positive second derivative at a point indicates concave up.
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Identify the concavity for $f(x) = x^3$ at $x = 0$.
Identify the concavity for $f(x) = x^3$ at $x = 0$.
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$f''(x) = 6x$, so $f''(0) = 0$. Concavity is undetermined. When $f''(x) = 0$, the second derivative test is inconclusive.
$f''(x) = 6x$, so $f''(0) = 0$. Concavity is undetermined. When $f''(x) = 0$, the second derivative test is inconclusive.
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What does it mean for a function to be concave down?
What does it mean for a function to be concave down?
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The graph is curved downwards, like a frown, and $f''(x) < 0$. Negative second derivative indicates downward curvature.
The graph is curved downwards, like a frown, and $f''(x) < 0$. Negative second derivative indicates downward curvature.
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What does it mean for a function to be concave up?
What does it mean for a function to be concave up?
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The graph is curved upwards, like a cup, and $f''(x) > 0$. Positive second derivative indicates upward curvature.
The graph is curved upwards, like a cup, and $f''(x) > 0$. Positive second derivative indicates upward curvature.
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