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Question: Let f(x) = \(\int_{1}^{x}{(2(x - 1)(x - 2)^{3} + 3(x - 1)^{2}(x - 2)^{2})dx}\), then-...

Let f(x) = ∫1x(2(x−1)(x−2)3+3(x−1)2(x−2)2)dx\int_{1}^{x}{(2(x - 1)(x - 2)^{3} + 3(x - 1)^{2}(x - 2)^{2})dx}, then-

A

f has exactly 4 critical points

B

f has maximum at x = 2

C

x = 75\frac{7}{5} is minima & x = 1 is maxima

D

None of these

Answer

x = 75\frac{7}{5} is minima & x = 1 is maxima

Explanation

Solution

f '(x) = 2(x – 1)(x – 2)3 + 3(x – 1)2(x – 2)2

= (x – 1)(x – 2)2(5x – 7) = 0

(for maxima and minima)

⇒ x = 1, 2, 75\frac{7}{5}

f(x) has maximum value at x = 1

minimum at x = 75\frac{7}{5}

and neither maxima & nor minima at x = 2