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Question Number 79876 by mathocean1 last updated on 28/Jan/20

$$fisderivablein\mathbb{R}.$$$$1)\mathrm{Demonstrate}\mathrm{that}\mathrm{if}f\mathrm{is}\mathrm{pair},f{}^{\prime }\mathrm{is}\mathrm{odd}\left(\mathrm{unpair}\right).$$$$1)\mathrm{Demonstrate}\mathrm{that}\mathrm{if}fis\mathrm{unpair},f{}^{\prime }\mathrm{is}\mathrm{pair}.$$$$$$$$\mathrm{Please}\mathrm{help}\mathrm{me}\mathrm{sirs}$$

Commented by MJS last updated on 29/Jan/20

$$\mathrm{if}‘‘{\mathrm{pair}}^{″}\mathrm{means}f(-x)=f\left(x\right)$$$$\mathrm{and}‘‘{\mathrm{unpair}}^{″}\mathrm{means}f(-x)=-f\left(x\right)$$$$\mathrm{then}\mathrm{i}.\mathrm{e}.$$$$y=\cos x\mathrm{and}y=x^2;y=x^4\mathrm{are}\mathrm{pair}$$$$y=\sin x\mathrm{and}y=x;y=x^3\mathrm{are}\mathrm{unpair}$$$$\mathrm{draw}\mathrm{them}\mathrm{and}\mathrm{try}\mathrm{to}\mathrm{understand}\mathrm{how}\mathrm{the}$$$$\mathrm{slopes}\mathrm{must}\mathrm{look}\mathrm{like}$$

Answered by Henri Boucatchou last updated on 29/Jan/20

$$fisderivableforall{x}_0\in \mathbb{R},{that}^{\prime }s$$$$\underset{x\to x_0}{lim}\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}=f{}^{\prime }\left(x_0\right)$$$$fpair\Rightarrow \underset{x-\to x_0}{lim}-\frac{f(-x)-f(-x_0)}{(-x)-(-x_0)}=-f{}^{\prime }(-x_0)=\underset{x\to x_0}{lim}\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}=f{}^{\prime }\left(x_0\right)$$$$\Rightarrow f{}^{\prime }(-x_0)=-f{}^{\prime }\left(x_0\right)e.i.f{}^{\prime }isodd;$$$$funpair\Rightarrow \underset{x\to x_0}{lim}-\frac{f(-x)-f(-x_0)}{(-x)-(-x_0)}=-f{}^{\prime }(-x_0)=\underset{x\to x_0}{lim}\frac{-[f\left(x\right)-f\left(x_0\right)]}{x-x_0}=-f{}^{\prime }\left(x_0\right)$$$$\Rightarrow -f{}^{\prime }(-x_0)=-f{}^{\prime }\left(x_0\right)$$$$\Rightarrow f{}^{\prime }(-x_0)=f{}^{\prime }\left(x_0\right)e.i.f{}^{\prime }ispair.$$

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