Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).
$ f(x) = \sin x $, $ 0 < x \leqslant \pi /2 $
Absolute maximum $f(\pi / 2)=1$. No local maximum. No local or absolute minimum.
we're going to sketch the graph of the function sine of X for X. Greater than syrup but less than or equal to play half. And we are going to use that sketch of the graph to find the absolute and local maximum and minimum values of the function. So we have this uh coordinate axis here and we know uh designed function has the maximum value of juan. Let's say yes at zero. We know it's zero but this is not included in the graph. So we're going to indicate that by um Yes. Bright. The function here again sign of eggs for X greater than syrup, but less and less than or equal to half. So we have that zero is not in the domain. So I draw this open circle here indicating that seriously included. Then we have the typical science shape function. Let's see here. It's not just like this but a little bit rounded near it sounds like this. And these values Uh corresponds to one at by heart and this is included. So indicate that with. Uh huh. Feel a red circle circle. So we can see in the graph that Thank you. There is no lowest point in the graph because the function is increasing all the time. All the way through the interval. From zero to buy health. So there is we cannot find a minimum value of the function on the whole into so F has no absolute minimum. And that's because execute seriously included. If we were included. The minimum value should be zero at zero but it's not included here. So there is no lowest flying into grass, it has no local max minimum for the same reason, there is not the point where we can say that is the lowest point and uh respect to absolute maximum we have f at by half is one and that's the highest point on the graph domain even here, So f by half equal one is the absolute maximum value Yeah, of F in or on over the interval Open zero and close by half, that is the value, one is the absolute maximum value of the function and it didn't attain or is of course at by half and we respect local maximum, there is no local maximum, so half, it has no local maximum because in this case is absolute maximum cannot be considered local because there is no function to write even the science to find there, it is not considering this case. So, these other properties of the function even here has no absolute minimum nor absolute local minimum, He has no local maximum, but it has an absolute maximum value of one which of course at X equals by half. That's the answer