The function y= x x^3 is strictly increasing function for

Step-by-step solution

Given the function y = x x^3 , we find its derivative with respect to x using the quotient rule. dy dx = x^3 d dx ( x) - x d dx (x^3) (x^3)^2 dy dx = x^3 1 x - x 3x^2 x^6 dy dx = x^2 - 3x^2 x x^6 dy dx = x^2(1 - 3 x) x^6 = 1 - 3 x x^4 For the function to be strictly increasing, we require dy dx > 0 . Since x^4 > 0 for all x > 0 , the condition simplifies to 1 - 3 x > 0 . 1 > 3 x x x Since the domain of x is x > 0 , the function is strictly increasing for 0 Answer: 0 < x < e^ 1 3