Problem 32 Easy Difficulty

If $ f'(x) = c $ ($ c $ a constant) for all $ x $, use Corollary 7 to show that $ f(x) = cx + d $ for some constant $ d $.

Answer

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Video Transcript

Okay, um we're being told that if f prime of X is equal to see and she is a constant for all X used Kerala very seven to show that f of X equals cfx. I mean, see explosive t with them constantly. So here, um in case you do not know what the Kerala very seven is and the Stuart textbook, I've restated it right here, and I'm going to stay there. So if f prime of X equals g prime back for all X and a B, then f of X equals G of X plus D for some constant d. So that is a statement. So we have to make a couple of assumptions. So we're told that F Prime of X is equal to see And in order to proceed with this, we're going to also create a function called G of X. So, So g of X, Um, we're gonna say it equals cfx so that when we take the derivative of G primary rex, it will equal See now, since primary, Rex and G Prime of X is equal to each other in this case, that and that's just a little case we made up. Um, we can actually apply Kerala very seven right away. Because corroborate 77 f prime lexical G prime of X. We can apply the statement f of X equals G of exports D So we can now. Now we can now use that formula f of X equals she of X Oh, dear. And since we have G FX, we can substitute that. You say after Becks is equal to see X plus d. Come on. And that is what we were asked. That's what we were being asked to prove so that of alphabetical cfx for the using corollary seven Oh!