4 for all values of x. How large can f(3) possibly be?EXAMPLE 5Suppose that f(0) = -2 and f'(x)We are given that f is differentiable (and therefore continuous) everywhere. In particular, we can apply theSOLUTIONMean Value Theorem on the interval O, 31. There exists a number c such thatro(-0)f(3) (0) f(c)Sof(3) f(0)f'(c)f'(c).=-2 +We are given that f'(x) s 4 for all x, so in particular we know that f'(c)Multiplying both sides of thisinequality by 3, we have 3f'(c)Sof(3) 2f(c) s -2+The largest possible value for f(3) is

Question
Asked Nov 5, 2019
4 for all values of x. How large can f(3) possibly be?
EXAMPLE 5Suppose that f(0) = -2 and f'(x)
We are given that f is differentiable (and therefore continuous) everywhere. In particular, we can apply the
SOLUTION
Mean Value Theorem on the interval O, 31. There exists a number c such that
ro(-0)
f(3) (0) f(c)
So
f(3) f(0)
f'(c)
f'(c).
=-2 +
We are given that f'(x) s 4 for all x, so in particular we know that f'(c)
Multiplying both sides of this
inequality by 3, we have 3f'(c)
So
f(3) 2
f(c) s -2+
The largest possible value for f(3) is
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4 for all values of x. How large can f(3) possibly be? EXAMPLE 5Suppose that f(0) = -2 and f'(x) We are given that f is differentiable (and therefore continuous) everywhere. In particular, we can apply the SOLUTION Mean Value Theorem on the interval O, 31. There exists a number c such that ro(-0) f(3) (0) f(c) So f(3) f(0) f'(c) f'(c). =-2 + We are given that f'(x) s 4 for all x, so in particular we know that f'(c) Multiplying both sides of this inequality by 3, we have 3f'(c) So f(3) 2 f(c) s -2+ The largest possible value for f(3) is

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Expert Answer

Step 1

Refer to the question we need to find the possible largest value of f(3) provided with the following information.

f(0)=-2
f'(x)4
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f(0)=-2 f'(x)4

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Step 2

Here apply the Mean value Theorem which states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a.b) then there is at least one number c in the interval (a,b) that is a < c < b such that,

f(b)-f(a)
f(c)
b-a
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f(b)-f(a) f(c) b-a

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Step 3

So us the above formula to ...

f(3)-(0)
(3-0)
- f'(c)
f(3)-f(0) (c)(3-0)
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f(3)-(0) (3-0) - f'(c) f(3)-f(0) (c)(3-0)

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Calculus