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

The point of inflation for a function is the point where the concavity changes. For this function, we can find the second derivative: $$f''(x) = 18xe^{-3x} - 18e^{-3x}$$ Setting this equal to zero and solving for x, we get: $$x = \frac{1}{3}$$ This is the point of inflation. To identify how it moves, we can check the sign of the second derivative for values of x around this point: For $x < \frac{1}{3}$, $f''(x) > 0$, so the function is concave up. For $x > \frac{1}{3}$, $f''(x) < 0$, so the function is concave down. Therefore, the point of inflation is moving closer to the origin as x approaches the point $\frac{1}{3}$ from either side (i.e. as x gets smaller or larger). Hence, the correct choice is C: point of inflation gets closer to the origin.

What Can You Say About Point of Inflation For f(x)=xe−3xf ( x ) = x e ^ { - 3 x }f(x)=xe−3x

What Can You Say About Point of Inflation For $f ( x ) = x e ^ { - 3 x }$