The Second Derivative Test utilizes concavity information to determine the type of critical point present. Since $p'(3)=0$ establishes a critical point and the function is concave up at $x=3$ (meaning $p''(3)>0$), the graph curves upward at this location. Upward concavity at a critical point creates a valley-shaped curve, confirming a local minimum. Choice A might attract students who confuse upward curvature with maximum behavior, but concave up specifically indicates the valley formation of minimums. Apply this test when both critical point and concavity direction are definitively known.

The Second Derivative Test determines the nature of critical points through concavity analysis. Since $G'(16)=0$ establishes a critical point and the function is concave up at $x=16$ (meaning $G''(16)>0$), the graph curves upward at this point. Upward concavity at a critical point forms a valley-shaped curve, indicating a local minimum. Choice B might confuse students who think upward curvature suggests maximums, but concave up specifically describes the valley formation of minimums. Apply this test when you can verify both critical point existence and the direction of concavity.

The Second Derivative Test examines concavity at critical points to determine their classification. Given $s'(20)=0$ (critical point) and $s''(20)>0$ (positive second derivative), the function is concave up at $x=20$. Positive concavity at a critical point produces upward curvature resembling a valley, confirming a local minimum. Choice C might tempt students who associate positive values with maximums, but positive second derivative specifically indicates the upward curvature of minimums. Use this test when both critical point status and second derivative sign are clearly determined.

The Second Derivative Test utilizes concavity information to classify critical points effectively. With $p'(13)=0$ confirming a critical point and $p''(13)<0$ indicating negative concavity, the function exhibits downward curvature at $x=13$. Downward concavity at a critical point creates a hill-like shape, establishing a local maximum. Choice A could mislead students who connect negative signs with minimums, but negative second derivative specifically refers to the hill formation characteristic of maximums. Ensure both conditions are met: critical point verification and determinable concavity sign.

The Second Derivative Test utilizes concavity information to classify critical points effectively. With $H'(-13)=0$ confirming a critical point and concave down behavior at $x=-13$ (meaning $H''(-13)<0$), the function exhibits downward curvature. Downward concavity at a critical point creates a hill-like shape, establishing a local maximum. Choice A could mislead students who connect downward direction with minimums, but concave down specifically refers to the hill formation characteristic of maximums. Ensure both conditions are met: critical point verification and determinable concavity sign.

The Second Derivative Test determines critical point nature through concavity analysis at those points. Since $p'(-6)=0$ establishes a critical point and $p''(-6)<0$ indicates negative concavity, the function is concave down at $x=-6$. Concave down behavior at a critical point forms a hill-shaped curve, indicating a local maximum. Choice A could confuse students who think negative signs correspond to minimums, but negative second derivative specifically means downward concavity (maximum). Apply this test when you have verified both the critical point condition and the sign of concavity.

The Second Derivative Test utilizes concavity information to classify critical points effectively. With $g'(-19)=0$ confirming a critical point and concave down behavior at $x=-19$ (meaning $g''(-19)<0$), the function exhibits downward curvature. Downward concavity at a critical point creates a hill-like shape, establishing a local maximum. Choice A could mislead students who connect downward direction with minimums, but concave down specifically refers to the hill formation characteristic of maximums. Ensure both conditions are met: critical point verification and determinable concavity sign.

This question tests understanding of the Second Derivative Test for classifying critical points. We have g'(-2) = 0, confirming a critical point exists at x = -2. The Second Derivative Test tells us that when f'(c) = 0 and f''(c) > 0, the function has a local minimum at x = c. Since g''(-2) = 7 > 0, the function is concave up at x = -2, indicating a local minimum. Choice B incorrectly suggests that having g''(-2) > 0 makes the result indeterminate, when in fact this positive value is exactly what allows us to classify the critical point. Remember: positive second derivative means concave up (local minimum), negative second derivative means concave down (local maximum).

This problem tests the Second Derivative Test for classifying critical points. Given q'(1) = 0, we have a critical point at x = 1. The Second Derivative Test tells us that when f'(c) = 0 and f''(c) < 0, the function has a local maximum at x = c. Since q''(1) = -9 < 0, the function is concave down at x = 1, which means we have a local maximum. Choice E incorrectly interprets q'(1) = 0 as meaning the function is constant, when it actually just means the tangent line is horizontal at that point. Remember the mnemonic: negative second derivative means the function "frowns" (concave down), creating a local maximum at the critical point.

This question requires applying the Second Derivative Test to determine the nature of a critical point. We have p'(5) = 0, confirming a critical point at x = 5. The Second Derivative Test states that when f'(c) = 0 and f''(c) > 0, the function has a local minimum at x = c. Since p''(5) = 12 > 0, the function is concave up at x = 5, indicating a local minimum. Choice A incorrectly identifies this as a point of inflection, which would require f''(c) = 0, not f'(c) = 0. When applying the Second Derivative Test, focus on the sign of f''(c): positive means the parabola opens upward (local minimum), negative means it opens downward (local maximum).