Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function, where $\mathbb{R}$ denotes the set of real numbers. Suppose that for all real numbers $x$ and $y$, the function $f$ satisfies $$f'(x) - f'(y) \leq 3|x - y|$$ Answer the following questions. No credit will be given without full justification. (a) Show that for all $x$ and $y$, we must have $\left|f(x) - f(y) - f'(y)(x - y)\right| \leq 1.5(x - y)^2$. (b) Find the largest and smallest possible values for $f''(x)$ under the given conditions.
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function, where $\mathbb{R}$ denotes the set of real numbers. Suppose that for all real numbers $x$ and $y$, the function $f$ satisfies
$$f'(x) - f'(y) \leq 3|x - y|$$
Answer the following questions. No credit will be given without full justification.\\
(a) Show that for all $x$ and $y$, we must have $\left|f(x) - f(y) - f'(y)(x - y)\right| \leq 1.5(x - y)^2$.\\
(b) Find the largest and smallest possible values for $f''(x)$ under the given conditions.