Eduprobe
Question:
If f(x) = ∫₀ˣ eᵗ²(t² - t³) dt for all x ∈ (0, ∞), then
f has a local maximum at x = 2
f has a local minimum at x = 3
f is decreasing on (2, 3)
there exists some c ∈ (0, ∞) such that f''(c) = 0
Solution:
f(x) = ∫₀ˣ eᵗ²(t² - t³) dt ∀x ∈ (0, ∞)
f'(x) has maxima at x = 2 (∵ f'(x) changes sign from +ve to -ve)
f'(x) has minima at x = 3 (∵ f'(x) changes sign from -ve to +ve). Also f(x) is decreasing in (2, 3) [∵ f'(x) < 0]
f'(x) = 0 for x = 2 and x = 3. So, by Rolle's theorem, there exists c ∈ (2, 3) for which f''(c) = 0. Option A, B, C and D are correct.