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We want to get the value of x that maximize $f ' (x) = \frac{4x^3}{3} - x^4$ . Let's calculate its derivate $f ''(x) = 4x^2 - 4x^3 = 4(x^2 - x^3) \Rightarrow f ''(0) = f '' (1) = 0$ ; (f '(1) > f '(0)) and $f '''(1) = 4(2 - 3) < 0 \Rightarrow \text{ f '(x) has a local maximum in x = 1}$ , furthemore $lim_{x \to \infty} f '(x) = - \infty \text{ and } lim_{x \to -\infty} f '(x) = - \infty \Rightarrow$ f '(x) has a global maximum at x = 1, because a global maximum would be also a local maximum in this case( f '(x) is infinitely derivative)