Suppose p(x) is a twice-differentiable function such that p(1) = 3, p'(1) = 0, and p”(1) = -2. Which of the following is true?
there is a relative maximum at p(1) = 3
there is a relative minimum at p(1) = 3
there is a point of inflection at p(1) = 3
none of these are true
p'(1)=0 so max or min
p"(1) < 0, so concave down: a max
(A)
Well, well, well, let's do some math magic! So, we have a function p(x) that is twice-differentiable, and we know its values and derivatives at x = 1. Now, buckle up and let's analyze!
Since p'(1) = 0, we know that the function p(x) has a horizontal tangent line at x = 1. Picture it like a flat pancake, no slopes, just a peaceful plain.
Now, let's move on to p''(1) = -2. This means that the concavity of the function changes at x = 1. It's like flipping the pancake and giving it a curvy shape. But does this mean we have a point of inflection at p(1) = 3? Oh, no no no!
Just because the concavity changes doesn't necessarily mean there's a point of inflection. It just means the function goes from being concave up to being concave down, or vice versa. It's like flipping a seesaw and changing the direction you're swinging!
So, we can conclude that none of the options are true. We don't have a relative maximum or a relative minimum at p(1) = 3, and there's no point of inflection either. It's just a peaceful, flat pancake with no extremes or inflections. Enjoy your pancake!
To determine whether there is a relative maximum, relative minimum, or point of inflection at p(1) = 3, we can use the values of the function and its derivatives at that point.
If p(1) = 3, it tells us the function value at x = 1.
If p'(1) = 0, it tells us that the slope of the tangent line at x = 1 is 0, which means the function is neither increasing nor decreasing at that point.
If p''(1) = -2, it tells us the concavity of the function at x = 1. A negative second derivative indicates concave down.
Based on this information, we can analyze the possibilities:
1. If there is a relative maximum at p(1) = 3, it means the function has a local maximum value at x = 1.
2. If there is a relative minimum at p(1) = 3, it means the function has a local minimum value at x = 1.
3. If there is a point of inflection at p(1) = 3, it means the function changes concavity at x = 1.
Using the given information, we find that p'(1) = 0 indicates there is a stationary point at x = 1, but it does not provide enough information to determine if it is a relative maximum or minimum. As p''(1) = -2 indicates concavity down, there is no point of inflection at x = 1.
Therefore, the correct answer is: none of these are true.
To determine whether there is a relative maximum, a relative minimum, or a point of inflection at p(1) = 3, we need to analyze the behavior of the function based on the given information about its derivatives.
A relative maximum or minimum occurs at a point where the derivative changes sign from positive to negative (relative maximum) or from negative to positive (relative minimum).
Let's analyze the given information step by step to determine the behavior at p(1)=3:
1. p(1) = 3: This tells us the value of the function at x=1, which is 3. However, it doesn't provide any information about the derivative at this point, so we cannot determine if it is a relative maximum or minimum yet.
2. p'(1) = 0: This tells us that the slope of the function at x=1 is 0. This point might have a relative maximum or minimum since the slope is changing from positive to negative or negative to positive. However, we need more information to make a conclusion.
3. p''(1) = -2: This tells us the second derivative of the function at x=1 is -2. The sign of the second derivative helps identify the concavity of the function at that point (positive: concave up, negative: concave down). A point of inflection occurs where the concavity changes.
Based on the given information, we have the following conclusions:
- The fact that p'(1) = 0 indicates the possibility of a relative maximum or minimum at x=1, but more information is needed.
- The fact that p''(1) = -2 indicates that the function is concave down at x=1.
To determine if there is a relative maximum or minimum, we need to examine the behavior of the function on either side of x=1. If the function is concave up before x=1 and concave down after x=1, it indicates a relative maximum. Conversely, if the function is concave down before x=1 and concave up after x=1, it indicates a relative minimum.
Since we don't have any additional information about the function or its behavior, we cannot make a definitive conclusion about whether there is a relative maximum or minimum at p(1) = 3. The correct answer is "none of these are true" because we don't have enough information to determine the behavior of the function at x=1.