Question

Let f(x) is a continuous function on [–7, 0] and differentiable function on the interval (–7, 0) such that f(–7) = –3 and given that f '(c) ≤ 2 for c ∈ (–7, 0). If the largest possible value of f(0) is k. Then the value of is ______.
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Answer 1

The values are [−∞, 20]

Step-by-step explanation:

Given: Let f(x) is a continuous function on [–7, 0] and differentiable function on the interval (–7, 0) such that f(–7) = –3 and given that f '(c) ≤ 2 for c ∈ (–7, 0). The largest possible value of f(0) is k.  

Find: The value

Solution:

f : [−7,0] → R

f(−7) = −3

f′(x) ≤ 2

Integrate →

f(x) ≤ 2x + c

Put x = −7

f(−7) ≤ −14 + c

c ≥ 14 − 3

c ≥ 11

f(x) ≤ 2x + 11

f(−1) ≤ 9

f(0) ≤ 11

f(0) + f(−1) ≤ 20

So the values are [−∞, 20]