2022-01-27

Finf the critical points of f(x,y)= e^xy +3 and use the second derivative test

alenahelenash

$f\left(x,y\right)={e}^{x}y+3$

Find the first derivative.

Since ${y}^{3}$ is constant with respect to $x$, the derivative of ${e}^{x}{y}^{3}$ with respect to $x$ is ${y}^{3}\frac{d}{dx}\left[{e}^{x}\right]$.

${f}^{\prime }\left(x\right)={y}^{3}\frac{d}{dx}\left({e}^{x}\right)$

Differentiate using the Exponential Rule which states that $\frac{d}{dx}\left[{a}^{x}\right]$ is ${a}^{2}\mathrm{ln}\left(a\right)$ where $a=e$

${f}^{\prime }\left(x\right)={y}^{3}{e}^{x}$

Find the second derivative.

Since ${y}^{3}$ is constant with respect to $x$, the derivative of ${e}^{x}{y}^{3}$ with respect to $x$ is ${y}^{3}\frac{d}{dx}\left[{e}^{x}\right]$.

${f}^{″}\left(x\right)={y}^{3}\frac{d}{dx}\left({e}^{x}\right)$

Differentiate using the Exponential Rule which states that $\frac{d}{dx}\left[{a}^{x}\right]$ is ${a}^{2}\mathrm{ln}\left(a\right)$ where $a=e$

${f}^{″}\left(x\right)={y}^{3}{e}^{x}$

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