$$\text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? }$$ Select the correct answer.
A) point of inflation gets closer to $\pm 6$
B) point of inflation goes away from the origin
C) point of inflation gets closer to the origin
D) point of inflation goes away from $\pm 3$

Question

Quizplus · Accepted Answer

To find the point of inflation, we need to differentiate the function and solve for when the derivative equals zero or does not exist.
$$f(x) = x e^{-3x}$$
$$f'(x) = e^{-3x} - 3xe^{-3x}$$
Setting this equal to zero and solving for x:
$$0 = e^{-3x}(1-3x)$$
$$e^{-3x}=0 \text{ or } 1-3x=0$$
The first equation has no solutions since $e^{-3x}$ is always positive. The second equation gives $x = \frac{1}{3}$.
Therefore, the point of inflation is at $x=\frac{1}{3}$. As $x$ increases, $f'(x)$ decreases since the exponential function decays faster than the linear function grows. This means that $f$ is concave down to the left of the point of inflation and concave up to the right of the point of inflation. As $x\rightarrow\pm\infty$, $f(x)\rightarrow 0$, so the point of inflation cannot be approaching $\pm\infty$. Therefore, the point of inflation must be getting closer to the origin, which eliminates choices A and D. The only remaining option is C, which is the correct answer.

What can you say about point of inflation for f(x)=xe−3x ? \text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? } What can you say about point of inflation for f(x)=xe−3x ?

$\text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? }$