Suppose that $ 3 \leqslant f'(x) \leqslant 5 $ for all values of $ x $. Show that $ 18 \leqslant f(8) - f(2) \leqslant 30 $.
Please see the explanation.
I suppose that three is less than a quarter have private access less than equal to five. For all values of X show that eighteen is less than equal effort. Bait minus effort to this lesson equals thirty. So we know from this equation that we are evaluating a number between and the interval from eight a home since I too could hate So we can actually write the mean value serum, Um, right off the top, because we know when and bees are so this will be f ate minus Q all over. Eight minus two. Typical at prime. See, Now we can simplify this further by writing rewriting this as half of eight minus efforts to equal to six times since we ate mine is too sick to be removed by six to the other side. We get sick. Time's a prime of C now. No, that we showed that that a six times after I'm back. We know that the interval from three the Western equals F prime of act um subside. Prime of eggs is less an equal of five. We have to multiply it both sides, all of it by six because we modified it right here. We multiplied the derivative at at sea, and we can think of this except C times six. So the most by this by six was by this side by six and multiply this side by sick. And so now we get this new wonderful statement, which I'm going to be right. Red six Emptory is eighteen. It's less than equal D'Oh, six times of crime. Uh, auf sie, which is percent aboutthe same thing on five times six thirty and since on six times after prime of sea is also equal to half of eight minus minus. Efforts to weaken re right inside. We can rewrite this. Actually, after three minus two. Forgot my little prentiss sister, it's less than you could have thirty. And this prove our statement. Disconnect. Um, that way showed that this this statement right here is actually also this statement as well. By the mean values D'oh