Concavity of Functions Over Their Domains - AP Calculus AB

1. Identify an inflection point for $f(x) = x^3 - 3x^2 + x$.

   Inflection at $x = 1$ where $f''(x)$ changes sign. Second derivative changes sign at this point.

2. Determine concavity of $f(x) = x^3 - x$ at $x = 1$.

   $f''(x) = 6x$, $f''(1) = 6 > 0$, concave up. Positive second derivative indicates upward concavity.

3. What does it mean for a function to be concave down?

   The graph is curved downwards, like a frown, and $f''(x) < 0$. Negative second derivative indicates downward curvature.

4. What is the concavity of $f(x) = \text{ln}(x)$ over $x > 0$?

   $f''(x) = -\frac{1}{x^2} < 0$, concave down for $x > 0$. Natural log is concave down on its domain.

5. Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$.

   $f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.

6. Identify concavity of $f(x) = \text{tan}(x)$ at $x = 0$.

   $f''(x) = 2\text{tan}(x)\text{sec}^2(x)$, $f''(0) = 0$. Undetermined. Zero second derivative requires further analysis.

7. What is the concavity of $f(x) = -x^2$ over its domain?

   $f''(x) = -2 < 0$, so $f(x)$ is concave down everywhere. Constant negative second derivative everywhere.

8. Identify concavity of $f(x) = \frac{1}{x^2}$ at $x = 1$.

   $f''(x) = \frac{6}{x^4}$, $f''(1) = 6 > 0$, concave up. Positive second derivative confirms upward concavity.

9. What is the concavity of $f(x) = x^5$ at $x = 0$?

   $f''(x) = 20x^3$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

10. Determine concavity of $f(x) = e^{-x}$ over its domain.

    $f''(x) = e^{-x} > 0$, so $f(x)$ is concave up everywhere. Exponential decay function maintains positive concavity.

11. Identify the concavity for $f(x) = \frac{1}{x}$ at $x = -1$.

    $f''(x) = \frac{2}{x^3}$, $f''(-1) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.

12. Identify concavity of $f(x) = \text{sin}(x)$ at $x = \text{π}$.

    $f''(x) = -\text{sin}(x)$, $f''(\text{π}) = 0$, undetermined. Zero second derivative makes concavity indeterminate.

13. Find the concavity of $f(x) = \text{ln}(x)$ at $x = 2$.

    $f''(x) = -\frac{1}{x^2}$, $f''(2) < 0$, concave down. Natural logarithm has negative second derivative.

14. What is the concavity of $f(x) = x^2$ over its domain?

    $f''(x) = 2 > 0$, so $f(x)$ is concave up everywhere. The second derivative is constant and positive.

15. What is the concavity of $f(x) = x^3 + 3x^2$ at $x = -1$?

    $f''(x) = 6x + 6$, $f''(-1) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

16. Determine concavity of $f(x) = \frac{1}{x}$ at $x = 1$.

    $f''(x) = \frac{2}{x^3}$, $f''(1) = 2 > 0$, concave up. Positive second derivative at a point indicates concave up.

17. How does the sign of $f''(x)$ affect the graph of $f(x)$?

    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.

18. What is the second derivative test for concavity?

    If $f''(x) > 0$, $f(x)$ is concave up; if $f''(x) < 0$, $f(x)$ is concave down. The second derivative determines the direction of curvature.

19. Identify the concavity for $f(x) = \text{cos}(x)$ at $x = \frac{\text{π}}{2}$.

    $f''(x) = -\text{cos}(x)$, $f''(\frac{\text{π}}{2}) = 0$, undetermined. When $f''(x) = 0$, concavity cannot be determined.

20. Identify concavity of $f(x) = \text{sin}(x)$ at $x = 0$.

    $f''(x) = -\text{sin}(x)$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

21. Identify the concavity for $f(x) = x^3$ at $x = 0$.

    $f''(x) = 6x$, so $f''(0) = 0$. Concavity is undetermined. When $f''(x) = 0$, the second derivative test is inconclusive.

22. Identify concavity of $f(x) = \text{arctan}(x)$ at $x = 0$.

    $f''(x) = -\frac{2x}{(1+x^2)^2}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.

23. What is the concavity of $f(x) = -x^3$ at $x = 1$?

    $f''(x) = -6x$, $f''(1) = -6 < 0$, concave down. Negative second derivative indicates downward concavity.

24. Determine concavity of $f(x) = x^2 - 4x$ at $x = 3$.

    $f''(x) = 2 > 0$, concave up. Constant positive second derivative for quadratic functions.

25. Identify concavity of $f(x) = \sin(x)$ at $x = \pi$.

    $f''(x) = -\sin(x)$, $f''(\pi) = 0$, undetermined. Zero second derivative makes concavity indeterminate.

26. At what point does concavity change for $f(x) = x^3$?

    Concavity changes at $x = 0$, where $f''(x) = 0$. Inflection occurs where $f''(x) = 0$ and changes sign.

27. Determine concavity of $f(x) = e^x$ over its domain.

    $f''(x) = e^x > 0$, so $f(x)$ is concave up everywhere. The exponential function has constant positive concavity.

28. Determine concavity of $f(x) = x^3 + 6x^2 + 9x$ at $x = -2$.

    $f''(x) = 6x + 12$, $f''(-2) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

29. Determine concavity of $f(x) = x^4 - 4x^2$ at $x = 1$.

    $f''(x) = 12x^2 - 8$, $f''(1) = 4 > 0$, concave up. Positive second derivative at the specified point.

30. What is the concavity of $f(x) = 4x^4 - x^2$ at $x = 0$?

    $f''(x) = 48x^2 - 2$, $f''(0) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.

31. Identify concavity of $f(x) = x^3 - 3x$ at $x = -1$.

    $f''(x) = 6x$, $f''(-1) = -6 < 0$, concave down. Negative second derivative indicates downward concavity.

32. What does it mean for a function to be concave up?

    The graph is curved upwards, like a cup, and $f''(x) > 0$. Positive second derivative indicates upward curvature.

33. Identify concavity of $f(x) = x^2 + 4$ at $x = 0$.

    $f''(x) = 2 > 0$, concave up. Constant positive second derivative for all quadratics.

34. What is the inflection point condition for $f(x)$?

    $f''(x)$ changes sign at an inflection point. Sign change in $f''(x)$ indicates a change in concavity.

35. What is the concavity of $f(x) = x^4$ at $x = 1$?

    $f''(x) = 12x^2$, $f''(1) = 12 > 0$, concave up. Positive second derivative indicates upward concavity.

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