Question

find the value of x at which the function has a possible relative maximum or minimum point. (recal for all x.) use the second derivative to determine the nature of the function at these points.

f(x)=\frac{1 - 10x}{e^{7x}}

what are the first and second derivatives of f(x)?

f(x)=\square

f(x)=\square

Explanation:

Step1: Find the first derivative using the quotient rule

The quotient rule is \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Let \(u = 1-10x\), so \(u^\prime=- 10\), and \(v = e^{7x}\), so \(v^\prime = 7e^{7x}\).

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Step2: Find the second derivative using the quotient rule

Now, for \(f^\prime(x)=\frac{70x - 17}{e^{7x}}\), let \(u = 70x-17\), \(u^\prime = 70\), and \(v = e^{7x}\), \(v^\prime=7e^{7x}\)

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Answer:

\(f^\prime(x)=\frac{70x - 17}{e^{7x}}\), \(f^{\prime\prime}(x)=\frac{189 - 490x}{e^{7x}}\)