Question

Question 7
If f'(x) > 3 for all x and f(2) = 5, what can you conclude about f(4)?
А
f(4) > 6
B
(4) > 8
С​

Answer 1

The value of f(4) is given by f(4) > 11

Correct question : If f'(x) > 3 for all x and f(2) = 5, what can you conclude about f(4) ? A. f(4) > 6 B. f(4) > 8 C. f(4) > 11 D. f(4) > 17

Given :

f'(x) > 3 for all x and f(2) = 5

To find :

The value of of f(4)

A. f(4) > 6

B. f(4) > 8

C. f(4) > 11

D. f(4) > 17

Solution :

Step 1 of 2 :

Write down the given function

Here the given function is f(x)

Now f'(x) > 3 for all x and f(2) = 5

Step 2 of 2 :

Find the value of f(4)

In the context of the problem Lagrange Mean value theorem states that ,

Let f be a real valued function defined on the closed interval [a,b] satisfying

i) f is continuous on the closed interval [a, b]

ii) f is differentiable on the open interval (a, b)

Then there exists a value x = c in (a, b) such that

$\displaystyle \sf{ f'(c) = \frac{ f(b) - f(a) }{b - a} }$

Now applying Lagrange Mean value Theorem for the given function on the interval [2,4] we get

$\displaystyle \sf{ f'(c) = \frac{ f(4) - f(2) }{4 - 2} }$

$\displaystyle \sf{ \implies \frac{ f(4) - 5 }{2} =f'(c) }$

$\displaystyle \sf{ \implies \frac{ f(4) - 5 }{2} > 3 \: \: \: \bigg[ \: \because \: f'(x) > 3 \: \forall \: x\bigg] }$

$\displaystyle \sf{ \implies f(4) - 5> 6}$

$\displaystyle \sf{ \implies f(4) > 11}$

Hence the correct option is C. f(4) > 11

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