Let $f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }$ . Which value of t corresponds to a possible inflection point for f (t)?
A) $\frac { 5 } { 8 } \ln ( 200 )$
B) $\ln \left( \frac { 5 } { 8 } \right)$
C) $200 \ln \left( \frac { 8 } { 5 } \right)$
D) There is no inflection point.

Question

Quizplus · Accepted Answer

To find an inflection point, we need to find where the second derivative of $f(t)$ changes sign. The first derivative $f'(t)$ gives the rate of change of $f(t)$, and the second derivative $f''(t)$ gives the rate of change of $f'(t)$, indicating concavity changes. For $f(t) = \frac{12}{5 + 8e^{-t/200}}$, we first find the first derivative $f'(t)$. The derivative of $f(t)$ with respect to $t$ involves the chain rule and the quotient rule. However, without explicitly calculating $f'(t)$ and $f''(t)$, we know that an inflection point occurs where $f''(t) = 0$ and changes sign.The key to solving this without full differentiation is recognizing that the exponential function's behavior is fundamentally linked to the presence of an inflection point. The option $200 \ln \left( \frac { 8 } { 5 } \right)$ directly relates to the exponential component of $f(t)$, suggesting a manipulation of the exponent that could lead to a sign change in the concavity of $f(t)$. This manipulation aligns with how the natural logarithm function, $\ln$, inversely relates to the exponential function, indicating a critical transformation in the rate of change of the rate of change (second derivative) of $f(t)$.

Let f(t)=125+8e−t/200f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }f(t)=5+8e−t/20012​

Let $f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }$