Expert answers to "Finf the critical points of f(x,y)= e^xy +3..."

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2022-01-27

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

alenahelenash

Expert2022-03-10Added 556 answers

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

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 $y^{3} \frac{d}{d x} [e^{x}]$ .

$f^{'} (x) = y^{3} \frac{d}{d x} (e^{x})$

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

$f^{'} (x) = y^{3} e^{x}$

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 $y^{3} \frac{d}{d x} [e^{x}]$ .

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

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

$f^{″} (x) = y^{3} e^{x}$

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