Let's compare the functions y equals x squared and y equals x cubed for positive x values. Looking at their graphs, we can see that these curves intersect at the point (1, 1). The blue curve represents x squared, and the red curve represents x cubed. Notice how their relative positions change depending on the x value.
To understand when x squared is less than x cubed, we need to analyze when x cubed minus x squared is positive. We can factor this expression as x squared times the quantity x minus 1. Since we're considering positive x values, x squared is always positive. Therefore, the sign of the difference depends entirely on whether x minus 1 is positive or negative. This occurs when x is greater than 1.
Now we can clearly see the three distinct regions. For x values between 0 and 1, x squared is greater than x cubed, meaning the blue curve is above the red curve. At x equals 1, both functions are equal. For x values greater than 1, x squared is less than x cubed, meaning the blue curve is below the red curve. This is why the original statement that x squared is lower than x cubed is only true when x is greater than 1.
Let's verify our analysis with specific numerical examples. When x equals 0.5, x squared equals 0.25 and x cubed equals 0.125. Since 0.25 is greater than 0.125, we confirm that x squared is higher than x cubed in this region. When x equals 2, x squared equals 4 and x cubed equals 8. Since 4 is less than 8, we confirm that x squared is lower than x cubed when x is greater than 1. These examples perfectly illustrate our mathematical analysis.
In conclusion, the statement that y equals x squared is lower than y equals x cubed is only partially correct. It's true specifically when x is greater than 1. For values between 0 and 1, the opposite is true. The intuition behind this is that when x is greater than 1, multiplying x squared by another factor of x, which is also greater than 1, results in a significantly larger number. This is why x cubed grows much faster than x squared for large values of x, making the cubic function eventually dominate the quadratic function.