KCET · Maths · Application of Derivatives
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 \(4\)
Answer & Solution
Correct Answer
(A) \(1\)
Step-by-step Solution
Detailed explanation
Given the function \(f(t) = -2t^3 + 3t^2 + 5\).
Differentiating with respect to \(t\), we get:
\(f'(t) = -6t^2 + 6t\)
For maximum or minimum, \(f'(t) = 0\):
\(-6t^2 + 6t = 0\)
\(-6t(t - 1) = 0\)
This gives \(t = 0\) or \(t = 1\).
Now, finding the second derivative:
\(f''(t) = -12t + 6\)
At \(t = 0\), \(f''(0) = 6 > 0\), which corresponds to a local minimum.
At \(t = 1\), \(f''(1) = -12(1) + 6 = -6 < 0\), which corresponds to a local maximum.
Thus, the video gets the maximum number of shares at \(t = 1\).
Answer: \(1\)
Differentiating with respect to \(t\), we get:
\(f'(t) = -6t^2 + 6t\)
For maximum or minimum, \(f'(t) = 0\):
\(-6t^2 + 6t = 0\)
\(-6t(t - 1) = 0\)
This gives \(t = 0\) or \(t = 1\).
Now, finding the second derivative:
\(f''(t) = -12t + 6\)
At \(t = 0\), \(f''(0) = 6 > 0\), which corresponds to a local minimum.
At \(t = 1\), \(f''(1) = -12(1) + 6 = -6 < 0\), which corresponds to a local maximum.
Thus, the video gets the maximum number of shares at \(t = 1\).
Answer: \(1\)
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