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Question Number 160606 by ZiYangLee last updated on 03/Dec/21

Evaluate (d^2 y/dx^2 ) when x=ln 2 and y=2.

$$\mathrm{Evaluate}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:\mathrm{when}\:{x}=\mathrm{ln}\:\mathrm{2}\:\mathrm{and}\:{y}=\mathrm{2}. \\ $$

Commented by MJS_new last updated on 03/Dec/21

y=f(x) (dy/dx)=f ′(x) (d^2 y/dx^2 )=f ′′(x) if y=2 ⇔ f(x)=2 ⇒ f(ln 2)=2 f(x)=2 ⇒ f ′(x)=0 ⇒ f ′′(x)=0 ⇔ (d^2 y/dx^2 )=0

$${y}={f}\left({x}\right) \\ $$$$\frac{{dy}}{{dx}}={f}\:'\left({x}\right) \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }={f}\:''\left({x}\right) \\ $$$$\mathrm{if}\:{y}=\mathrm{2}\:\Leftrightarrow\:{f}\left({x}\right)=\mathrm{2}\:\Rightarrow\:{f}\left(\mathrm{ln}\:\mathrm{2}\right)=\mathrm{2} \\ $$$${f}\left({x}\right)=\mathrm{2}\:\Rightarrow\:{f}\:'\left({x}\right)=\mathrm{0}\:\Rightarrow\:{f}\:''\left({x}\right)=\mathrm{0}\:\Leftrightarrow\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\mathrm{0} \\ $$

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