(c, \( f(c) \) ) on the graph of \( f \) bl al point, (c) both function \( f \) \( f^{\prime \prime} \) exist on on \( (a, b) f \) is \( a, b \) ), then , (c) concave for which \( f^{\prime} \) and \( f \) function suppose \( c \), en the there is a lo differentiable on ction \( f \) is differ er \( f^{\prime \prime}(c)=0 \) then \( (c, f(c)) \) is an inflection in Problems 27-34, an object in rectilinear motion moves along a 25. \( f(\theta)=\sin \theta-2 \cos \theta \) 20. \( f(x)=x+2 \sin x \) horisontal line with the positive direction to the right. The distances difi object from the origin at time \( t \geq 0 \) (in seconds) is given. (a) Determine the intervals during which the object moves to the right and the intervals during which it moves to the left. (b) When does the object reverse direction? (c) When is the velocity of the object increasing and when is it decreasing? (d) Draw a figure to illustrate the motion of the object. (c) Draw a figure to illustrate the velocity of the object. 27. \( s=t^{2}-2 t+3 \) 28. \( s=2 t^{2}+8 t-7 \) 29. \( s=2 t^{3}+6 t^{2}-18 t+1 \) 30. \( s=3 t^{4}-16 t^{3}+24 t^{2} \) 39. \( f(x)=2 x^{3}-6 x^{3}+6 x \) 41. \( f(x)=x^{4}-4 x \) 43. \( f(x)=5 x^{4}-x^{5} \) 45. \( f(x)=3 x^{5}-20 x^{3} \) 47. \( f(x)=x^{2} e^{x} \) 120 49. \( f(x)=6 x^{4 / 3}-3 x^{1 / 3} \) (314) 51. \( f(x)=x^{2 / 3}\left(x^{2}-8\right) \) 53. \( f(x)=x^{2}-\ln x \) 55. \( f(x)=\frac{x}{\left(1+x^{2}\right)^{5 / 2}} \) 57. \( f(x)=x^{2} \sqrt{1-x^{2}} \) (12) 59. \( f(x)=x-2 \sin x \).

Hello! It looks like you've shared a series of calculus and motion-related problems involving functions, their derivatives, concavity, inflection points, and applications to rectilinear motion. To provide the most effective assistance, could you please specify which particular problem or concept you'd like help with? In the meantime, I can offer a general overview of some of the key concepts mentioned: ### **1. Inflection Points and Concavity** An **inflection point** on the graph of a function \( f \) is a point \( (c, f(c)) \) where the concavity of the function changes. To determine if a point is an inflection point: - **First Derivative (\( f' \))**: Provides information about the slope and increasing/decreasing behavior of the function. - **Second Derivative (\( f'' \))**: Indicates the concavity of the function. - If \( f''(c) = 0 \) and changes sign around \( c \), then \( (c, f(c)) \) is an inflection point. - If \( f''(c) \) does not change sign, then \( c \) is not an inflection point. ### **2. Analyzing Rectilinear Motion** For an object moving along a straight line, the position \( s(t) \) as a function of time \( t \) can be analyzed to determine: - **Direction of Motion**: - **To the Right (Positive Direction)**: When the velocity \( v(t) = s'(t) > 0 \). - **To the Left (Negative Direction)**: When \( v(t) = s'(t) < 0 \). - **Reversing Direction**: Occurs when the velocity changes sign, i.e., \( v(t) = 0 \). - **Acceleration**: - **Increasing Velocity**: When acceleration \( a(t) = v'(t) > 0 \). - **Decreasing Velocity**: When \( a(t) = v'(t) < 0 \). ### **3. Example Problem Analysis** Let's briefly analyze **Problem 27** as an example: **Problem 27:** \( s(t) = t^2 - 2t + 3 \) **a) Determine the intervals during which the object moves to the right and to the left.** - **Velocity**: \( v(t) = s'(t) = 2t - 2 \) - **Set Velocity to Zero**: \( 2t - 2 = 0 \) → \( t = 1 \) - **Test Intervals**: - For \( t < 1 \): Choose \( t = 0 \), \( v(0) = -2 < 0 \) → Moves to the **left**. - For \( t > 1 \): Choose \( t = 2 \), \( v(2) = 2 > 0 \) → Moves to the **right**. **b) When does the object reverse direction?** - At \( t = 1 \), since the velocity changes from negative to positive. **c) When is the velocity of the object increasing and when is it decreasing?** - **Acceleration**: \( a(t) = v'(t) = 2 \) - Since \( a(t) = 2 > 0 \) for all \( t \), the velocity is **increasing** for all \( t \). **d) Draw a figure to illustrate the motion of the object.** While I can't draw here, you can sketch the position function \( s(t) = t^2 - 2t + 3 \), which is a parabola opening upwards with its vertex at \( t = 1 \). The velocity graph \( v(t) = 2t - 2 \) is a straight line crossing the \( t \)-axis at \( t = 1 \). --- If you have a specific problem from your list you'd like to delve into or need clarification on a particular concept, feel free to let me know!