Let the function f:−$$7$$,$$0$$→R be continuous on −$$7$$,$$0and$$ differentiable on −$$7$$,$$0$$. If f−$$7=$$−$$3and$$ f'x≤$$2$$ for all such function f, f−$$1+f0$$ is in the interval:

Question

Let the function  f:−$$7$$,$$0$$→R be continuous on  −$$7$$,$$0and$$ differentiable on −$$7$$,$$0$$. If  f−$$7=$$−$$3and$$  f'x≤$$2$$ for all such function f, f−$$1+f0$$ is in the interval:

Infinity Learn · Accepted Answer

Lets use LMVT for x∈−$$7$$,−$$1$$  f−$$1$$−f−$$7$$−$$1+7=f$$'x≤$$2$$ ⇒f−$$1+36$$≤$$2$$ ⇒f−$$1$$≤$$9$$   →$$1$$                                    Also use LMVT forx∈−$$7$$,$$0$$  $$f0$$−f−$$70+7=f$$'x≤$$2$$⇒$$f0+37$$≤$$2$$ ⇒$$f0$$≤$$11$$   →$$2$$ Adding $$1$$ and $$2$$, we get            ∴$$f0+f$$−$$1$$≤$$20$$

Let the function $f : [- 7, 0] \to R$ be continuous on $[- 7, 0]$ and differentiable on $(- 7, 0)$ . If $f (- 7) = - 3$ and $f^{'} (x) \leq 2$ for all such function f, $f (- 1) + f (0)$ is in the interval:

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Let the function f:−7,0→R be continuous on −7,0and differentiable on −7,0. If f−7=−3and f'x≤2 for all such function f, f−1+f0 is in the interval:

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Let the function $f : [- 7, 0] \to R$ be continuous on $[- 7, 0]$ and differentiable on $(- 7, 0)$ . If $f (- 7) = - 3$ and $f^{'} (x) \leq 2$ for all such function f, $f (- 1) + f (0)$ is in the interval: