Let \( N(t) = 2t - t^4 \) and \( D(t) = (1 + t^3)^2 \), then:
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- I'(t) = rac{u'v - uv'}{v^2} = rac{(2t)(1 + t^3) - t^2(3t^2)}{(1 + t^3)^2}.
- I'(t) = rac{2t + 2t^4 - 3t^4}{(1 + t^3)^2} = rac{2t - t^4}{(1 + t^3)^2}.
- Now, find \( I''(t) \) to locate where \( I'(t) \) has a maximum:
- I''(t) = rac{N'(t)D(t) - N(t)D'(t)}{[D(t)]^2}.
- Compute derivatives:
- N'(t) = 2 - 4t^3, \quad D'(t) = 2(1 + t^3)(3t^2) = 6t^2(1 + t^3).
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