Question

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

f(x)=(5x - 4)e^{1 - 2x}

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

f(x)=

Explanation:

Step1: Find the first derivative using the product rule

The product rule is \((uv)^\prime = u^\prime v+uv^\prime\). Let \(u = 5x - 4\) and \(v=e^{1 - 2x}\).
\(u^\prime=5\) and \(v^\prime=- 2e^{1 - 2x}\)

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Step2: Find the second derivative using the product rule

Let \(u = 13 - 10x\) and \(v = e^{1 - 2x}\). Then \(u^\prime=-10\) and \(v^\prime=-2e^{1 - 2x}\)

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Answer:

\(f^\prime(x)=e^{1 - 2x}(13 - 10x)\)
\(f^{\prime\prime}(x)=e^{1 - 2x}(20x - 36)\)