A.𝑓(𝑡)=2𝑡−𝑡3f(t)=2 t −t 3 B.𝑑(𝑡)=(1.1)𝑡d(t)=(1.1) t C.𝑏(𝑡)=𝑡4−3𝑡+9b(t)=t 4 −3t+9D.ℎ(𝑡)=5𝑡+𝑡5h(t)=5 t +t 5 E.𝑐(𝑡)=𝑡2−5𝑡c(t)= t 2 −5t​

To analyze the functions provided, I will break down each function into smaller parts, apply relevant concepts, analyze the functions in detail, and provide a concise summary of findings.

1. Break Down the Problem

The functions are polynomial expressions defined with respect to $t$:

• A. $f(t) = 2t - t^3$
• B. $d(t) = (1.1)^t$
• C. $b(t) = t^4 - 3t + 9$
• D. $h(t) = 5t + t^5$
• E. $c(t) = t^2 - 5t$

2. Relevant Concepts

The primary concepts involved include:

• Polynomial functions and their properties (degree, leading coefficient, roots).
• Exponential functions, specifically analyzing growth rates.
• Derivative and critical points for understanding behavior.

3. Analysis and Detail

1. Function Analysis:

   • A. $f(t) = 2t - t^3$:

     • Degree is 3; the leading coefficient is negative, indicating a downwards opening parabola.
     • Critical points found by setting $f'(t) = 2 - 3t^2 = 0$.
     • Roots can be found by solving $t^3 - 2t = 0$.

   • B. $d(t) = (1.1)^t$:

     • An exponential growth function, growing indefinitely as $t$ approaches infinity.

   • C. $b(t) = t^4 - 3t + 9$:

     • Degree is 4; leading coefficient is positive, meaning it opens upwards.
     • Find critical points to determine maximum and minimum.

   • D. $h(t) = 5t + t^5$:

     • Degree is 5; leading coefficient is positive, thus it opens upwards.
     • Analyze behaviors for large values of $t$.

   • E. $c(t) = t^2 - 5t$:

     • Quadratic function, forming a parabola that opens upwards.
     • Roots can be found using the quadratic formula.

4. Verify and Summarize

Each function serves different mathematical behaviors and future implications based on growth, decay, and critical points:

• A. Has points of inflection and local maxima/minima; typically analyzed for optimization problems.
• B. Shows exponential growth with no upper bound.
• C. Involves higher degree influences, used in more complex mathematical modeling.
• D. Similar to B and C but with the introduction of both linear and polynomial behavior.
• E. Quadratic nature reveals symmetry and specific vertex behavior.

Final Answer

The functions are polynomial or exponential with distinct properties influencing their behavior based on their degree. $f(t), b(t), h(t),$ and $c(t)$ have polynomial characteristics, while $d(t)$ exhibits exponential growth. Each function's analysis can help predict behavior under certain conditions (e.g., maximums/minimums, growth rates).