Question

A Youtube short video is getting viral according to $f(t) = -2t^3 + 3t^2 + 5$. At what time does the video get maximum number of shares? (t is in hours)

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For simple polynomial optimization, finding the roots of the first derivative usually points directly to the answer. Always perform a quick mental check of the second derivative (or the sign change of the first derivative) to ensure you've found a maximum and not a minimum.

Answer 1

The Correct Option is A

1. To find the time at which the video gets the maximum number of shares, we need to find the critical points of the function \( f(t) = -2t^3 + 3t^2 + 5 \). This involves finding the derivative of the function and setting it equal to zero.
2. First, compute the derivative of \( f(t) \) with respect to \( t \):

\(f'(t) = \frac{d}{dt}(-2t^3 + 3t^2 + 5)\) \(= -6t^2 + 6t\)

1. Set the derivative \( f'(t) \) to zero to find the critical points:

\(-6t^2 + 6t = 0\)

Factor the equation:

\(6t(t - 1) = 0\)

This gives two solutions: \( t = 0 \) and \( t = 1 \).

1. To determine which of these points gives a maximum, we can use the second derivative test. Compute the second derivative:

\(f''(t) = \frac{d}{dt}(-6t^2 + 6t)\) \(= -12t + 6\)

1. Evaluate the second derivative at each critical point:
   • At \( t = 0 \): \(f''(0) = -12 \cdot 0 + 6 = 6\), which is positive. This indicates a local minimum.
   • At \( t = 1 \): \(f''(1) = -12 \cdot 1 + 6 = -6\), which is negative. This indicates a local maximum.
2. Thus, the video gets the maximum number of shares at \( t = 1 \) hour.

Therefore, the correct answer is option \(1\).