Let f: [-1, 2] rightarrow R be a differentiab | vedclass

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(A) Given $0 \le f'(t) \le 1$ for $t \in [-1, 0]$.Integrating with respect to $t$ from $-1$ to $0$:$\int_{-1}^{0} 0 \, dt \le \int_{-1}^{0} f'(t) \, d

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$-2 \le f(2) - f(-1) \le 1$

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(A) Given $0 \le f'(t) \le 1$ for $t \in [-1, 0]$.Integrating with respect to $t$ from $-1$ to $0$:$\int_{-1}^{0} 0 \, dt \le \int_{-1}^{0} f'(t) \, dt \le \int_{-1}^{0} 1 \, dt$$0 \le f(0) - f(-1) \le 1$ ... $(1)$Given $-1 \le f'(t) \le 0$ for $t \in [0, 2]$.Integrating with respect to $t$ from $0$ to $2$:$\int_{0}^{2} -1 \, dt \le \int_{0}^{2} f'(t) \, dt \le \int_{0}^{2} 0 \, dt$$-2 \le f(2) - f(0) \le 0$ ... $(2)$Adding inequalities $(1)$ and $(2)$:$(0 - 2) \le (f(0) - f(-1)) + (f(2) - f(0)) \le (1 + 0)$$-2 \le f(2) - f(-1) \le 1$.

Let $f: [-1, 2] \rightarrow R$ be a differentiable function such that $0 \le f'(t) \le 1$ for $t \in [-1, 0]$ and $-1 \le f'(t) \le 0$ for $t \in [0, 2]$. Then:

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