Okay, I need major help! Can someone tell me if these statements are true or false ASAP please. Thank you.

1. If ƒ′(x) < 0 when x < c then ƒ(x) is decreasing when x < c.

True

2. The function ƒ(x) = x^3 – 3x + 2 is increasing on the interval -1 < x < 1.

False

3. If ƒ'(c) < 0 then ƒ(x) is decreasing and the graph of ƒ(x) is concave down when x = c.

True

4.A local extreme point of a polynomial function ƒ(x) can only occur when ƒ′(x) = 0.

True

5. If ƒ′(x) > 0 when x < c and ƒ′(x) < 0 when x > c, then ƒ(x) has a maximum value when x = c.

False

6. If ƒ′(x) has a minimum value at x = c, then the graph of ƒ(x) has a point of inflection at x = c.

False

7. If ƒ′(c) > 0 and ƒ″(c), then ƒ(x) is increasing and the graph is concave up when x = c.

True

8. If ƒ′(c) = 0 then ƒ(x) must have a local extreme point at x = c.

False

9. The graph of ƒ(x) has an inflection point at x = c so ƒ′(x) has a maximum or minimum value at x = c.

True

10. ƒ′(x) is increasing when x < c and decreasing when x > c so the graph of ƒ(x) has an inflection point at x = c.

True

So yeah those are the questions (statements), if you can tell which ones are true / false or correct me on what I said that would be helpful. I put what I thought it was so if its wrong, please correct me! Thanks,

Veronica!!

im from the future and currently taking this exact online quiz. "S", I hope you know youre lame! People need things like this now more than ever. Thank you.

1. T agree - if x is increasing, f(x) is decreasing

2. F agree - max at x = -sqrt 3, min at x = +sqrt 3
3. False, disagree, it is decreasing but who says it is concave or convex? It could be a straight line with negative slope.

4.A local extreme point of a polynomial function ƒ(x) can only occur when ƒ′(x) = 0.

True AGREE

5. If ƒ′(x) > 0 when x < c and ƒ′(x) < 0 when x > c, then ƒ(x) has a maximum value when x = c.

TRUE I think - DISAGREE looks like /\

6. If ƒ′(x) has a minimum value at x = c, then the graph of ƒ(x) has a point of inflection at x = c.

False
TRUE - DISAGREE
an extreme value of f' means f" changes sign which means inflection

Agree with 7 8 9 10

oh okay, thanks so much! :) i really appreciate the help, especially since u corrected what i did wrong. thanks!

Not only that, but you are wrong on number 6.

A minimum value of f'(x) at x = c does NOT imply an extreme value, although the reverse is certainly true if the sign changes from - to +.

f''(x) could be undefined at f'(c) which would make it a critical value of f'(x) since it is in the domain of f'(x). An example would be if f'(x) = x ^ (1/2).

(0,0) is clearly a minimum value of f'(x), and in fact f(x), but it is not a relative minimum so there would be no inflection point.

Let's go through each statement one by one and determine if it is true or false:

1. If ƒ′(x) < 0 when x < c then ƒ(x) is decreasing when x < c.
This statement is true. If the derivative of a function is negative for all x-values less than c, then the function itself is decreasing on that interval.

2. The function ƒ(x) = x^3 – 3x + 2 is increasing on the interval -1 < x < 1.
This statement is false. To determine if a function is increasing or decreasing on an interval, you need to find the derivative of the function and check its sign. In this case, the derivative is ƒ'(x) = 3x^2 - 3. When you substitute values between -1 and 1, you get 3(-1)^2 - 3 = 0, which means the derivative is zero and the function does not have a consistent sign, so it is neither increasing nor decreasing on the given interval.

3. If ƒ'(c) < 0 then ƒ(x) is decreasing and the graph of ƒ(x) is concave down when x = c.
This statement is true. If the derivative of a function is negative at a specific point c, it means the function is decreasing at that point. Additionally, if the second derivative is negative at that point, it means the graph of the function is concave down.

4. A local extreme point of a polynomial function ƒ(x) can only occur when ƒ′(x) = 0.
This statement is true. Local extreme points of a function occur where the derivative is either zero or undefined. So, if ƒ'(x) = 0 at a point, it is a potential local extreme point.

5. If ƒ′(x) > 0 when x < c and ƒ′(x) < 0 when x > c, then ƒ(x) has a maximum value when x = c.
This statement is false. If ƒ′(x) changes sign from positive to negative at x = c, it indicates a local maximum. However, it doesn't guarantee that the function has a maximum value at that point. It could still have a minimum or neither.

6. If ƒ′(x) has a minimum value at x = c, then the graph of ƒ(x) has a point of inflection at x = c.
This statement is false. A minimum value of ƒ'(x) at x = c indicates that the slope of the graph of ƒ(x) is smallest at that point. It does not necessarily indicate a point of inflection.

7. If ƒ′(c) > 0 and ƒ″(c) > 0, then ƒ(x) is increasing and the graph is concave up when x = c.
This statement is true. If both the first and second derivatives are positive at x = c, it means the function is increasing, and the graph is concave up at that point.

8. If ƒ′(c) = 0, then ƒ(x) must have a local extreme point at x = c.
This statement is false. ƒ'(c) = 0 means the derivative is zero at x = c, which implies a potential local extreme point, but it does not guarantee it. It could be a point of inflection instead.

9. The graph of ƒ(x) has an inflection point at x = c, so ƒ′(x) has a maximum or minimum value at x = c.
This statement is true. At an inflection point, the derivative ƒ'(x) can have a maximum or minimum value.

10. ƒ′(x) is increasing when x < c and decreasing when x > c, so the graph of ƒ(x) has an inflection point at x = c.
This statement is true. When the derivative ƒ'(x) changes sign from increasing to decreasing at x = c, it indicates that the graph of ƒ(x) has an inflection point at that location.

I hope this helps clarify the true and false statements. If you have any further questions, feel free to ask!

I hope you realize by doing this you are helping students taking online calculus cheat on their test, as this is direct content from the test under Unit 6 Activity 9.