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Calculus/differential
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Question
could you give me the reason for choosing f'(x)=0;if f(X)is a real function?and also could you give me the proof of having
f'(C)=f(b)-f(a)/b-a in mean value theorem
The reason for choosing f'(x)=0 if for finding the extreme points of the function since when an extreme point is reached, continuous functions go from a slope up to a slope down, making it 0 at the extreme point.
The mean value theorem assumes the function and its derivatives are both continuous over the interval [a,b] and the C is some point in between a and b. f'(C) is the slope of a straight line from (a,f(a)) to (b,f(b)).
Let's suppose that at the start the graph slopes up faster at a than the average slope. To come into b at f(b), the function needs to have a lesser slope than the average somewhere. Since the derivative is assumed to be continuous, then at some point in the interval the slope needs to be the average slope since it was greater at one end and lesser at the other.
If the graph slopes up slower than the average change at the start, at some point in the middle it must slope up faster to reach (b,f(b)). It must therefore (from continuity) be equal the average slope somewhere in the middle.
If the graph slope is equal to the average slope at the left side, we're done before we started.
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