Final answer:
While evaluating the expressions 3x and 6x^2, it is correct that for any value of x greater than 1, 6x^2 will be larger than 3x, due to the nature of quadratic growth versus linear growth. However, for negative values of x, the inequality still holds since squaring a negative number yields a positive result, and thus 6x^2 will also be larger than 3x.
Step-by-step explanation:
When evaluating the claim that 6x^2 will always take larger values than 3x for the same values of x, we must consider the behavior of each function. For x > 0, 6x^2 grows much faster than 3x because it is a quadratic function, whereas 3x is linear. In particular, as x increases, the square of x (which is x^2) increases at an accelerated rate compared to x itself. This behavior is confirmed by the rules of exponents, where multiplying an exponential expression by itself amplifies the growth rate extensively.
For x = 1, both expressions are equal (6*1^2 = 6, and 3*1 = 6). However, for any value of x greater than 1, 6x^2 will indeed be larger than 3x. For example, at x = 2, 6x^2 yields 6*2^2 = 24, while 3x yields 3*2 = 6. The claim holds in this case. Nevertheless, it's crucial to note that for x < 0 (negative values of x), the claim does not hold because the values of 3x will be negative while those of 6x^2 will still be positive due to the squaring of a negative number resulting in a positive number.
We can generalize that 6x^2 will always be greater than 3x for all x > 1, and for negative x, 6x^2 is also greater. Therefore, Lin's conclusion is largely correct, especially for positive values of x greater than 1.