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 \leqslant x < \pi /2 $
No absolute or local
maximum. Absolute minimum $f(0)=0$. No local
we are going to sketch the graph of the function sine of X. For X greater than or equal to zero. And lessons I have. And is that graph to find the absolute and local maximum and minimum values of the function. So we know that the sine function At zero is equal to zero and then it is increasing from zero to buy half that by healthy. It has a value one but in this case is not included in the graph. So we can say that we have something like this. Okay. And this corresponds to by half here mm Value one. She'll be a little bit more flatten around here but it's not probably Okay. So we can say that the by the one should be around and mm should be around here. Everything right? So to indicate and describe that the image of my half Which is one. Normally it's not included because the enjoyed something. We have this and here at zero is included. So we do this type of graphic sign to indicate that. Yeah. So we can see that F. That's the formula to remember. We're talking about sign of eggs. For eggs. Okay, there we go to zero but less than by half. And so we have effectively we have uh absolute minimum at zero and the value zero. So F zero equals zero is the absolute minimum of minimum value of F on zero. I have zero. Close by. Have opened is the absolute minimum there's no relative minimum because this is a satirical Syria cannot be considered local because we have no uh function to the left, even though the sign is defined, we have considered only on that internal, so there is no local minimum, have has no local meaningful on that interval. Uh We respected the absolute maximum of local maximum, there is neither one of them because we have opened this interval here, there is, there is not a maximum or highest point on the graph because the point is not good. So if has no local or absolute maximum. And so this is the behavior as a function sine of X for X greater than or equal to zero, but less less than by half. There is an absolute minimum value of zero at 0 and there is no local minimum, no local maximum, no absolute maximum.