Explain why or why not:
1. if f'(c)=0, then f has a local maximum or minimum at c.
2. if f''(c)=0, then f has an inflection point at c.
3. F'(x)=x^2+10 and G(x)=x^2-100 are antiderivatives of the same function
4. Between two local minima of a function continuous on (-∞,∞), there must be a local minimum.
#1. only if f" ≠ 0. Consider f(x) = x^3
#2. only if f' ≠ 0. Consider f(x) = x^4
#3. I guess so, if F" = G'
If you meant to type F(x) instead of F'(x), then I'd have to say true.
#4. f has a local min when f' changes from negative to positive.
So, if f(x) has minima at x=a,b with a < b, then for some some small h,
f'(a-h) < 0
f'(a+h) > 0
f'(b-h) < 0
f'(b+h) > 0
So, at some point c such that a+h < c < b-h we must have f'(c) = 0 and thus f"(c) < 0, meaning f(c) is a local max.
But what if f' is not continuous? It could switch from + to - without having to be 0 at x=c.
So, what about
f(x) =
x - x^2/2 + 3/2 for x <= 3/2
2x - x^2/2 for x > 3/2
the limit as x→3/2 is 3, so f(x) is continuous, but f'(x) is discontinuous at x = 3/2. f(x) as a minimum at both x=1,2 but no maximum in between.
If you strengthen the condition to inckude that both f and f' are continuous, then the statement is true.
I mistyped #4, it should be:
Between two local minima of a function continuous on (-∞,∞), there must be a local MAXIMUM.
1. Well, if f'(c) = 0, it means that the slope of the function is zero at point c. However, this doesn't necessarily mean that f has a local maximum or minimum at c. It could just be a point of inflection, or even a plateau. So, in conclusion, your statement is about as accurate as trying to find a needle in a haystack with a blindfold on.
2. Ah, the second question. If f''(c) = 0, it means that the second derivative of the function is zero at point c. This indicates a possible inflection point, but it's not a guarantee. There are cases where f''(c) = 0 and the function doesn't have an inflection point. So, it's not the ultimate proof you were hoping for. It's more like finding a pot of gold at the end of the rainbow – it's possible, but don't count on it.
3. Oh, the exciting world of antiderivatives! If F'(x) = x^2 + 10 and G(x) = x^2 - 100 are antiderivatives of the same function, then it means that the derivative of F(x) is x^2 + 10, and the derivative of G(x) is also x^2 + 10. So, even though F(x) and G(x) look different, they're still secret twins. They're like two peas in a pod, but with different hairstyles.
4. Ah, the universe of local minima. Between two local minima of a function continuous on (-∞,∞), there must be a local minimum. It's like a mandatory pit stop for all functions on their road trip. If a function reaches a bottom point, it can't just zoom past another bottom point – it has to stop and say hello. So, it's like a little pitstop, a teeny tiny vacation spot for the function. Just enough time for it to catch its breath before continuing on its merry way.
1. If f'(c) = 0, it means that the derivative of the function f(x) is zero at the point c. However, it does not guarantee that f has a local maximum or minimum at c. In order to determine if there is a local maximum or minimum, we would need additional information such as the sign of the derivative around c. If the derivative changes signs from negative to positive around c, then f has a local minimum at c. If the derivative changes signs from positive to negative around c, then f has a local maximum at c. Otherwise, if the derivative is zero but does not change signs, it does not necessarily indicate the presence of a local maximum or minimum.
2. If f''(c) = 0, it means that the second derivative of the function f(x) is zero at the point c. However, a point where the second derivative is zero does not guarantee the presence of an inflection point. An inflection point occurs when the concavity of a function changes, and this can only be determined by analyzing the behavior of the function on both sides of c, including the behavior of the first and second derivatives. Therefore, although the second derivative being zero is a necessary condition for an inflection point, it is not sufficient. Additional analysis of the function is required to determine if there is a genuine inflection point at c.
3. To determine if F'(x) = x^2 + 10 and G(x) = x^2 - 100 are antiderivatives of the same function, we need to compare the derivatives of F(x) and G(x). Taking the derivative of F(x) gives us F'(x) = 2x, while the derivative of G(x) is G'(x) = 2x. Since the derivatives are equal, it implies that F(x) and G(x) differ only by a constant. Therefore, F(x) and G(x) are indeed antiderivatives of the same function.
4. The statement is not true. Between two local minima of a function continuous on (-∞, ∞), there can be other types of critical points, such as local maxima or points of inflection. The presence of a local minimum between two local minima depends on the specific behavior of the function in that region and cannot be guaranteed solely based on the presence of local minima.
1. If f'(c) = 0, it means that the derivative of the function f at the point c equals zero. However, this does not guarantee that f has a local maximum or minimum at c. In fact, f(c) could be a point of inflection or an endpoint of a function, in which case f does not have a local maximum or minimum at c. To determine if f has a local maximum or minimum at c, we need to use the second derivative test or analyze the behavior of the function around c.
To use the second derivative test, we calculate f''(c). If f''(c) > 0, then f has a local minimum at c. If f''(c) < 0, then f has a local maximum at c. If f''(c) = 0, we cannot determine if there is a local maximum or minimum at c.
2. If f''(c) = 0, it means that the second derivative of the function f at the point c equals zero. Having f''(c) = 0 does not automatically indicate that f has an inflection point at c. An inflection point occurs when the concavity of a function changes. To determine if f has an inflection point at c, we need to examine the concavity of the function around c by analyzing the sign changes of f''(x) in the neighborhood of c.
If f''(x) changes sign from positive to negative (or negative to positive) around c, then f has an inflection point at c. If the sign of f''(x) does not change around c, there is no inflection point at c.
3. To determine if F'(x) = x^2 + 10 and G(x) = x^2 - 100 are antiderivatives of the same function, we need to check if their derivatives are equivalent.
Let's find the derivatives of F(x) and G(x):
F'(x) = d/dx(x^2 + 10) = 2x
G'(x) = d/dx(x^2 - 100) = 2x
Since F'(x) = G'(x) = 2x for all x, it indicates that F(x) and G(x) differ only by a constant value. Therefore, F(x) and G(x) are indeed antiderivatives of the same function.
4. If a function is continuous on the interval (-∞, ∞) and has at least two local minima, then there must also be a local minimum between those two minima. This is because the function is continuous, meaning that it has no breaks or jumps in its graph. In order for a local minimum to exist, the function must have a value lower than all other points in its immediate neighborhood. Therefore, if there are two local minima, there must be a continuous segment connecting them, and within that segment, there will be a local minimum.