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Suppose that f : R → R is a continuous function on the interval [-3, 3] and a differentiable function in the interval (-3, 3) such that for every x in the interval, f'(x) ≤ 2. If f(-3) = 7, then f(3) is at most _______.

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Best answer

Answer:19 to 19

Data

f: R -> R 

continuous in [-3.3]

differentiable in (-3,3) 

f(-3) = 7

 f'(x) < = 2

Calculation:

=>f'(x) < = 2

Integrating both side from -3 to 3

 \(\mathop \smallint \limits_{ - 3}^3 f'\left( x \right)dx <= 2\mathop \smallint \limits_{ - 3}^31dx\)

[f(3)-f(-3)] <= 2[3-(-3)]

f(3)<= 7 + 2(6)

f(3)<=19

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