00:01
So in this problem we have that f of x is greater than 0 for all x and g of x is 1 over f of x.
00:12
So now in part a we have that f is increasing on the interval i, on some interval i around x not.
00:23
That means that for any x and y inside the interval i, we have that y greater than x implies is f of y is greater than equal to f of x that's what it means to say that if is increasing and that implies that now f of y is one over g y and f of x is one over g x so from this inequality we have that one over g of y is greater than equal to one over g of x and now f y f x g y f x are all positive so now we can take the reciprocal of this inequality.
01:13
And we have g of y is less than equal to g of x.
01:22
That is, g is decreasing in the interval i around x now.
01:35
Now next in part b we have that f has a local maximum at some point x1.
01:45
And that means that there exists some epsilon greater than zero such that for all x in the interval x1 minus epsilon comma x1 plus epsilon so we are taking an interval of a small interval around x1 and in that interval we have f of x1 is greater than equal to f of x or all x in that interval and that implies that 1 over g of x1 because remember g is 1 over f so f is 1 over g of x 1 is greater than equal to 1 over g of x is greater than equal to 1 over g of x again all these are positive quantities so we can take the reciprocal and we have or we can like cross multiply whatever and we get the inequality g of x 1 is less than equal to g of x.
02:57
That is, g has a local minimum at x1.
03:05
In part c, we have that f is concave down at x2.
03:13
That is, there exists an interval, say, a, comma b around x2, such that, or all x and y in that interval a comma b and for any lambda in the open interval zero comma one we have that f of one minus lambda x plus lambda y is greater than equal to one minus lambda times f of x plus lambda times f of y that's the definition of concave down or simply concave.
03:50
Now by the weighted amgm inequality we have for any lambda in the open interval 0 comma 1, 1 minus lambda times gy plus lambda times g of x is greater than equal to g of y raised to the power 1 minus lambda times g of x raised to the power lambda...