Stationary Points

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Find the derivative of $f(x) = \frac{e^{x^{2}} \ln(x)}{x}$ and then use it to locate all stationary points of $f$ .

f'(x) = \frac{e^{x^{2}}\big((2x^{2}+1)\ln x + 1\big)}{x^{2}}

f'(x) = \frac{e^{x^{2}}\big((x^{2}-1)\ln x + 1\big)}{x^{2}}

f'(x) = \frac{e^{x^{2}}\big((2x^{2}-1)\ln x - 1\big)}{x^{2}}

f'(x) = \frac{e^{x^{2}}\big((2x^{2}-1)\ln x + 1\big)}{x^{2}}

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A company's profit function is modeled by $P(x) = (x^2 - 4x + 3)e^{2x}$ thousand dollars, where $x$ is the number of units produced (in hundreds). Use differentiation to find $P'(x)$ and then determine the stationary point(s) of the profit function (where $P'(x)=0$ ). Enter $P'(x)$ as your answer.

(2x - 4)e^{2x} - 2(x^2 - 4x + 3)e^{2x}

P'(x) = 2e^{2x}(x^2 - 3x + 1)

thousand dollars per hundred units

(2x - 4)e^{2x} + (x^2 - 4x + 3)e^{2x}

(2x - 4)e^{2x}

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Consider the function $f(x) = x \ln(x) - x$ . To locate its critical points, first compute $f'(x)$ .

\ln x

0

\ln x + 1

\frac{1}{x} - 1

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Stationary Points

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