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Consider a function g(x) which is defined and differentiable on (-8,8) and increasing in (1,2) and decreasing elsewhere. We construct another function f(x) = g(x) - (g(x))^2 + (g(x))^3. Find domain of f(x), it’s interval of monotonicity.
f(x) = g(x) - (g(x))^2 + (g(x))^3f'(x) = g'(x) ( 1 - 2 g(x) + 3g(x)^2)Now, 3g(x)^2 - 2g(x) + 1 has Discriminant = 0. Hence, this means that this is either always positive or always negative.g'(x) > 0 in the interval (1,2)g'(x) < 0 in the interval (-8,1) U (2,8)The domain of f(x) is same as that of g(x) which is (-8,8).For the interval of monotonicity are :We cannot clearly say that 3g(x)^2 - 2g(x) + 1 is positive or negative as we do not know much about g(x). Say it is positive. Hence,f(x) is monotonically increasing in interval (1,2)f(x) is monotically decreasing in interval (-8,1) U (2,8)ThanksBharat BajajIIT Delhiaskiitians faculty
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