Q:

The graph of f’’(x) is continuous and decreasing with an x-intercept at x=-3. Which of the following statements must be true?A. The graph of f is always concave downB. The graph of f has an inflection point at x=-3 C. The graph of f has a relative minimum at x=-3D. None of these are true

Accepted Solution

A:
I cannot say that I am entirely sure of the answer so let me know if it doesn't make sense, but I will try to explain as best as I can nonetheless.1. The graph of f''(x) represents the graph of the second derivative of f(x). Now, we know that the graph is continuous and decreasing. I think that the most important thing here is to mentally visualise the graph - if it is decreasing and has an x-intercept at x = -3, then we can say the following:a) for all values of x before -3, f''(x) is positiveb) at x = -3, f''(x) is 0c) for all values of x after x = -3, f''(x) is negative2. What this means in terms of the graph f'(x) is the following:a) for values of x less than -3, the gradient of the graph of f'(x) is positive and becoming less positive as x reaches 0b) at x = -3, the gradient of the graph of f'(x) is 0c) for values of x more than -3, the gradient of the graph of f'(x) is negative and becoming more negative as x reaches ∞ With this in mind, maybe try drawing a quick sketch to guide you (I would include one here but I have trouble adding attachments so I hope you'll forgive my lack of one) - it could perhaps look something similar to -(x + 3)^2 (but wouldn't be restricted to this - remember, it is just a visual aid).3. Now, we need to work from the graph of f'(x) to the graph of f(x).What we need to notice is that the graph of f'(x) takes the form of a concave down graph - this means that the gradient of the graph of f(x) immediately to either side of x = -3 changes from being either:a) + >> ++ >> +++ >> ++ >> +(Here, the number of + symbols signifies the strength of the positive gradient. >> represents an arrow.So, the gradient starts off less positive, becomes more positives, reaches its peak, and then becomes gradually less positive again - imagine this being represented by f'(x) = -(x + 3)^2 + 5 (again, remember this is just a visual aid) )b) --- >> -- >> - >> -- >> ---(Likewise, the number of - symbols signifies the strength of the negative gradient.So, the gradient starts off very negative, becomes less negative, reaches its peak, and then gradually becomes more negative again - you can see that this is effectively the same pattern as above: there is an increasing trend and then a decreasing trend. You can imagine this as being represented by the graph f'(x) = -(x + 3)^2 - 5)c) -- >> - >> 0 >> - >> --(Here, the gradient is negative, becomes less negative, reaches 0, then gradually becomes more negative - again, there is the same increasing trend followed by a decreasing trend. You can imagine this as being represented by the graph f'(x) = -(x + 3)^2)It is this increasing trend in the gradient up to x = -3 followed by a decreasing trend that is crucial to take note of - this signifies that there is a point of inflection at x = -3. What we must remember here is that a point of inflection is characterised by a change in the curvature of the graph - either from concave up to concave down or from concave down to concave up. In our case, this would be a transition from concave up to concave down as the gradient gradually becomes more positive until it reaches its highest value at x = -3 and then gradually becomes less positive. Thus, we can say that answer B (the graph of f has an inflection point at x = -3) is correct.Looking at the other answers:A - The graph of f cannot be always concave down since there is a clear change in the gradient from less positive to more positive to less positive again (if it were always concave down the gradient would just gradually become more negative)C - A relative minimum is characterised by the fact that the gradient to the left of the minimum is negative, the gradient at the minimum is 0, and the gradient to the right of the minimum is positive. Since this isn't the case for our graph, this is not the correct answer.D - This is only a viable answer if none of the others are correct; since we have identified B as correct, this is incorrect.I hope this helped but if you have any questions or problems with my working, please don't hesitate to comment below.