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6. The graph of the A. On what interval(s) is / derivative f of a function increasing? fis given below on (0,6).C On what interval(s) is f concave up Or concave dow...

Question

6. The graph of the A. On what interval(s) is / derivative f of a function increasing? fis given below on (0,6).C On what interval(s) is f concave up Or concave down?fB: At what value(s) of x does have Iocal maximum?D: At whit value(s) of x does have an inflection point?

6. The graph of the A. On what interval(s) is / derivative f of a function increasing? fis given below on (0,6). C On what interval(s) is f concave up Or concave down? f B: At what value(s) of x does have Iocal maximum? D: At whit value(s) of x does have an inflection point?



Answers

$(a)$ the intervals on which $f$ increases and the intervals on which $f$ decreases; (b) the local maxima and the local minima; (c) the intervals on which the graph is concave up and the intervals on which the graph is concave down; (d) the points of inflection. Use this information to sketch the graph of $f.$ $$f(x)=x^{1 / 3}(x-6)^{2 / 3}$$

So we're gonna want to sketch this graph but I suggest that you also graph um the equation um f prime of X equals x squared plus six minus x or minus six. So ffx equals x squared 1st X -6. I'm doing this. We see that negative three and two are both going to be this one right here is going to be since it's increasing and then decreasing. This one is going to be a maximum points. So something like this and this one is going to be a minimum points or something like this, but this is also going to be a cubic function. So what we'll see is that we'll have an inflection point that occurs right at this point right here. So this is going to be an inflection point where it goes from concave down two concave uh So that would be an important way of viewing the graph. And as we see, this is still going to be the cubic function that we expected.

Define problem. We see the graph of f. Prime is shown. So we want to know on what intervals is it decreasing? So let's say we're given um a function such as three X squared my next to the function that looks like this. And we want to know where the function is increasing or decreasing on what intervals. So you see it's going to be increasing when the derivative is positive. Now this isn't the graph that you're given, but it's going to have very similar applications. Shall we see that the funk that the positive function is right here from here to negative infinity. So we see from negative infinity, Choose negative 0.816 And then from 0.816 to infinity, that's gonna be when the functions increasing, but it's gonna be decreasing in between here. And then we see that a local minimum will be reached um Since this can be increasing and then decreasing, this would be a local maximum, but this will be a local minimum.

Define problem. We see the graph of f. Prime is shown. So we want to know on what intervals is it decreasing? So let's say we're given um a function such as three X squared my next to the function that looks like this. And we want to know where the function is increasing or decreasing on what intervals. So you see it's going to be increasing when the derivative is positive. Now this isn't the graph that you're given, but it's going to have very similar applications. Shall we see that the funk that the positive function is right here from here to negative infinity. So we see from negative infinity, Choose negative 0.816 And then from 0.816 to infinity, that's gonna be when the functions increasing, but it's gonna be decreasing in between here. And then we see that a local minimum will be reached um Since this can be increasing and then decreasing, this would be a local maximum, but this will be a local minimum.

Were given the graph of the derivative s crime of a function f This is in the figure of exercise five in part they were asked to find on the intervals on which F is increasing or decreasing. So, looking at our graph, we see that the derivative F prime of X is greater than zero on the open interval 15 therefore and we see that f prime changes from negative to positive I'm sorry so far is that f is increasing on 15 We also see that F prime of X is less than zero on the open air balls 01 and 56 So it follows that the function F is decreasing on the open intervals 01 and 56 in part B were asked to find the values of X, at which the function F has a local maximum or a minimum. So to do this, essentially be applying the first derivative rule. Notice that F prime of X equals zero add X equals one, and also that F prime changes from negative to positive at X equals one. Therefore, by the first derivative test, F has a local minimum at X equals one. Likewise, we see that F prime of X is equal to zero at X equals five and F prime changes from positive to negative at X equals five. So it follows that by the first derivative test, F has a local maximum at X equals five.


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