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Numerade Educator

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Problem 33 Medium Difficulty

Let $ f(x) = 1/x $ and
$$ g(x) = \left\{
\begin{array}{ll}
\frac{1}{x} & \mbox{ if} x > 0\\
1 + \frac{1}{x} & \mbox{ if} x < 0
\end{array} \right. $$
Show that $ f'(x) = g'(x) $ for all $ x $ in their domains. Can we conclude from Corollary 7 that $ f - g $ is constant?

Answer

For $x>0, f(x)=g(x),$ so $f^{\prime}(x)=g^{\prime}(x) .$ For $x<0, f^{\prime}(x)=(1 / x)^{\prime}=-1 / x^{2}$ and $g^{\prime}(x)=(1+1 / x)^{\prime}=-1 / x^{2},$ so
again $\left.f^{\prime}(x)=g^{\prime}(x) . \text { However, the domain of } g(x) \text { is not an interval [it is }(-\infty, 0) \cup(0, \infty)\right]$ so we cannot conclude that
$f-g$ is constant (in fact it is not).

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Video Transcript

Okay, so we're told toe every back equals one of the wrecks. And what to vex. Be apiece wise to find function if f X rated Indio was one of the first lessons. Zero is one plus one back. And then we're being asked to show that if f prime of ex show that f crime Mexico's G prime back for all accent of remain. And then the question after us, can we conclude from Kerala every seven, that f my energy is constant. So we're going to go ahead and first through the statement from Mexico to Private Rex. So since X when X is greater than zero, we know that after Mexico G events, because when your ex is great and you know it's one of the wrecks and FX is one of wreck. So we know that FX equal to Jax, and then we can also conclude at F Primex. It's also Ecology Private X for ex lessons. You know, um, we know that effort. Becks is one of Rex, and it's derivative deal with the ex is is one over X. Off the neck comes our negative one over X squared and then for G of X g prime of X. It's just a derivative of one plus one over X. I'm sorry about that one over X, which is also negative. One over X quit. So again we proven that at primary backs isn't it equal to G prime attacks? No less proven. Now the question is, can we conclude from Kerala? Very seven. That f manages a concert. And so if you recall or karate every seven years, it's asked if effort if prime Rex ical g prime break for all acts in an interval So in an interval, Um then we can say that every vaccine Koji breakfast for some constant d. So first of all, the big problem here is that we don't have an interval for g avec. So no interval interval for Jia vex. So we can't really make We can't really apply crawling seven without a certain interval. So technically we cannot conclude that f minor Jia's content. So no, it may be, but we can't We can't assume that with what we're given, so no