The problem involves analyzing the function $f(x) = (0.5x - 1.45)e^{2x}$ and determining its behavior, including its minimum point, based on the provided variation table.

Key Points in the Table:

1. Sign of $f'(x)$:

   • Negative ($-$) for $x \in (-\infty, 2.4)$
   • Zero ($0$) at $x = 2.4$
   • Positive ($+$) for $x \in (2.4, +\infty)$

   This indicates that the function $f(x)$:

   • Decreases on $(-\infty, 2.4)$
   • Reaches a minimum at $x = 2.4$
   • Increases on $(2.4, +\infty)$

2. Value of $f(x)$ at $x = 2.4$:

   • The function's value at the minimum is given as $f(2.4) = -30.378$.

3. Conclusion from the Table:

   • The function $f(x)$ has a minimum at $x = 2.4$, and the value of the minimum is $-30.378$.

Final Answer:

• $f$ presents a minimum at $x = 2.4$, and the minimum value is $-30.378$.

Do you want me to:

1. Explain how to compute $f(x)$ or $f'(x)$?
2. Elaborate on how the signs of $f'(x)$ relate to the behavior of $f(x)$?
3. Solve for the derivative of $f(x)$ step by step?
4. Clarify why $f(2.4)$ is the minimum value?
5. Discuss similar problems for practice?

Tip:

Always verify the critical points of a function by checking where the derivative equals zero and analyzing the intervals around those points to confirm the function's behavior.