Question

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

f(x)=(5x - 3)e^{3-6x}

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

Explanation:

Step1: Find the first derivative

Use the product rule \((uv)^\prime = u^\prime v+uv^\prime\) where \(u = 5x - 3\), \(v=e^{3 - 6x}\).
\(u^\prime=5\), \(v^\prime=- 6e^{3 - 6x}\)

$$
LATEXBLOCK0
$$

Step2: Find the second derivative

Use the product rule again, where \(u=-30x + 23\), \(v = e^{3 - 6x}\)
\(u^\prime=-30\), \(v^\prime=-6e^{3 - 6x}\)

$$
LATEXBLOCK1
$$

Answer:

\(f^\prime(x)=e^{3 - 6x}(-30x + 23)\)
\(f^{\prime\prime}(x)=e^{3 - 6x}(180x - 168)\)