Trials graph question (1 Viewer)

[BlueGas]

I need help doing these type of questions.

AND this question too

 

[dan964]

I need help doing these type of questions.

AND this question too

1. Do you know what f'(a)<0 or f'(a)>0 and f''(a)>0 or f''(a)<0 means. The first is the slope of the tangent at a, the second is the concavity (which is harder to determine its value just off the graph, but we can tell whether it is positive or negative). (Used a instead of alpha)
2. Note that the area below the graph (consider it negative) cancels the area of the top of the graph. So calculate the area above the graph and this must equal the area below the graph for it to be zero.

Thus the triangle below the graph must equal that value, and hence we can find the base of the triangle (the height is given), and hence a.

 

[BlueGas]

1. Do you know what f'(a)<0 or f'(a)>0 and f''(a)>0 or f''(a)<0 means. The first is the slope of the tangent at a, the second is the concavity (which is harder to determine its value just off the graph, but we can tell whether it is positive or negative). (Used a instead of alpha)
2. Note that the area below the graph (consider it negative) cancels the area of the top of the graph. So calculate the area above the graph and this must equal the area below the graph for it to be zero.

Thus the triangle below the graph must equal that value, and hence we can find the base of the triangle (the height is given), and hence a.

f''(a)<0 means it's a maximum
f''(a)>0 means it's a minimum

but how about the first two?

 

[keepLooking]

f''(a)<0 means it's a maximum
f''(a)>0 means it's a minimum

but how about the first two?

They are the gradients of the f(x) graph. Which in this case, f(a) < 0

 

[Mikes]

6. B: the graph is clearly decreasing at x=a, giving us a negative gradient function, ie., f'(a)<0. Also, since the graph is concave up at x=a the second derivative gives a positive, ie., f''(a)>0.

9. D: the value of 'a' must be 15 so that the area of the triangle is 30 units^2. (remember that the integral for the area under the x axis comes out as a negative)

Do you have the correct answers?

 

[Flop21]

For the first picture / question.

f'(a) = function is decreasing or increasing

f''(a) = concavity i.e. concave up or concave down.


So by looking at the graph, it's decreasing (moving in the negative direction), and it's concave up (on the part of the curve that is concave up).

Therefore, f'(a) < 0 [because it's decreasing] and f''(a) > 0 (concave up).

 

[BlueGas]

6. B: the graph is clearly decreasing at x=a, giving us a negative gradient function, ie., f'(a)<0. Also, since the graph is concave up at x=a the second derivative gives a positive, ie., f''(a)>0.

9. D: the value of 'a' must be 15 so that the area of the triangle is 30 units^2. (remember that the integral for the area under the x axis comes out as a negative)

Do you have the correct answers?

Yes, they are the correct answers.

 

[dan964]

f''(a)<0 means it's a maximum
f''(a)>0 means it's a minimum

but how about the first two?

nope, that only applies if it is a stationary point which it is isn't.
f'(a) means the gradient , so f'(a) >0 means gradient (slope) is positive, and vice versa
f''(a) is concavity, so f''(a) > 0, means concave up (U-shaped), and f''(a)<0 means concave-down

see Flop21 post for an explanation and Mike for the actual answers.

 

[dan964]

They are the gradients of the f(x) graph. Which in this case, f(a) < 0

I get what you are trying to say, but the notation is f'(a) not f(a).
The value of the gradient has nothing to do with whether f(a) is bigger or less than 0.