Final answer:
To prove the uniform continuity of a function on a closed interval, it suffices to show that the function is continuous on that interval. The function in question is a piecewise polynomial, which is continuous, and upon applying the relevant theorem, it can be concluded that the function is uniformly continuous on the interval [−2,1].
Step-by-step explanation:
The student has asked to prove that the function f(x) is uniformly continuous on the interval [−2,1]. To prove this, we can apply a theorem which states that if a function is continuous on a closed interval, then it is also uniformly continuous on that interval. The function given can be broken into two parts: when x ≤0, f(x) = 0, and when x > 0, f(x) = −x³ + 7x² + 2x. Both parts are polynomials, which are continuous everywhere, thus f(x) is continuous in each piece of the defined interval. Since the interval [−2,1] is closed and bounded, f(x) is continuous on the interval. Therefore, by the theorem, f(x) is uniformly continuous on [−2,1].