Draw the graph of following: y = x 3 - 4 x 4

Given, y = f x = x 3 - 4 x 4 Clearly, f ( x ) is a polynomial function of a degree 3 , since the domain of and polynomial function is R . So, the domain of f ( x ) is R . Range of f ( x ) will be R since it is an odd degree polynomial function. For, f ( x ) = 0 ⇒ x 3 - 4 x 4 = 0 ⇒ x ( x 2 - 4 ) 4 = 0 ⇒ x = 0 , - 2 , 2 [point of intersection of f x with x - axis] So, at x = 0 , f ( 0 ) = 0 (where passes through the origin) For monotonic behaviour find f ' ( x ) ⇒ f ' x = 3 x 2 - 4 4 Foe finding critical points, we must have f ' ( x ) = 0 ⇒ 3 x 2 - 4 4 = 0 ⇒ x = ± 2 3 [critical points] [sign of f ' ( x ) ] So, f x is monotonically increasing in - ∞ , - 2 3 ∪ 2 3 , ∞ and, f x is monotonically decreasing in - 2 3 , 2 3 For the point of inflection f '' ( x ) = 0 , f '' x = 6 x 4 ⇒ f '' x = 0 ⇒ 6 x 4 = 0 ⇒ x = 0 [point of inflection] [sign of f '' ( x ) ] f x is concave upward for x > 0 . f x is concave downward for x < 0 . Putting critical points in f '' ( x ) for maxima and minima ⇒ f ' ' - 2 3 = - 6 2 3 < 0 And, f '' 2 3 = 6 2 3 > 0 Therefore, x = 2 3 is a point of minima and x = - 2 3 is a point of maxima. As, x → - ∞ , lim x → - ∞ f ( x ) = lim x → - ∞ x 3 - 4 x 4 → - ∞ x → ∞ , lim x → ∞ f ( x ) = lim x → ∞ x 3 - 4 x 4 → ∞