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

Step 1: Understanding the Concept:
We need to find the time $t$ at which a given function $f(t)$, representing the number of shares, reaches its maximum value. This is a classic optimization problem solvable using the first and second derivative tests.

Step 2: Key Formula or Approach:
1. Find the first derivative $f'(t)$. 2. Set $f'(t) = 0$ to find the critical points (potential maxima or minima). 3. Find the second derivative $f''(t)$. 4. Evaluate $f''(t)$ at the critical points. A negative value indicates a local maximum.

Step 3: Detailed Explanation:
The function for the number of shares is: \[ f(t) = -2t^3 + 3t^2 + 5 \] Calculate the first derivative with respect to $t$: \[ f'(t) = \frac{d}{dt}(-2t^3 + 3t^2 + 5) = -6t^2 + 6t \] To find critical points, set the first derivative to zero: \[ -6t^2 + 6t = 0 \] Factor the equation: \[ -6t(t - 1) = 0 \] This gives two critical points: $t = 0$ and $t = 1$. Since time $t$ must be meaningful in the context of the problem (usually $t>0$ for something "getting viral" after posting), $t=1$ is our primary candidate. Let's verify it's a maximum. Calculate the second derivative: \[ f''(t) = \frac{d}{dt}(-6t^2 + 6t) = -12t + 6 \] Evaluate the second derivative at the critical points: At $t = 0$: $f''(0) = -12(0) + 6 = 6>0$. This indicates a local minimum. At $t = 1$: $f''(1) = -12(1) + 6 = -6<0$. Since the second derivative is negative, the function attains a local maximum at this point. Therefore, the maximum number of shares occurs at $t = 1$ hour.

Step 4: Final Answer:
The video gets the maximum number of shares at $t = 1$.