Suppose that f is differentiable for all x and that f prime( x ) ≤ 2 for all x. If f(1)=2 and f (4)=8 then f(2) has the value equal to

Question

Suppose that f is differentiable for all x and that f prime( x ) ≤ 2 for all x. If f(1)=2 and f (4)=8 then f(2) has the value equal to (A) 3 (B) 4 (C) 6 (D) 8

Tardigrade Admin · Accepted Answer

Using LMVT for f in [1,2] ∀ c ∈(1,2) ( f (2)- f (1)/2-1)= f prime( c ) ≤ 2 f (2)- f (1) ≤ 2 ⇒ f (2) ≤ 4....(1) again using LMVT in [2,4] ∀ d ∈(2,4) ( f (4)- f (2)/4-2)= f prime( d ) ≤ 2 ∴ f (4)- f (2) ≤ 4 8- f (2) ≤ 4 4 ≤ f (2) f (2)=4....(2) from (1) and (2) f (2) = 4

Q. Suppose that f is differentiable for all x and that f′(x)≤2 for all x. If f(1)=2 and f(4)=8 then f(2) has the value equal to

3

4

6

8

Using LMVT for f in [1,2] ∀c∈(1,2)2−1f(2)−f(1)​=f′(c)≤2 f(2)−f(1)≤2⇒f(2)≤4....(1) again using LMVT in [2,4] ∀d∈(2,4)4−2f(4)−f(2)​=f′(d)≤2 ∴f(4)−f(2)≤4 8−f(2)≤4 4≤f(2)f(2)=4....(2) from (1) and(2)f(2)=4

Q. Suppose that $f$ is differentiable for all $x$ and that $f^{'} (x) \leq 2$ for all $x$ . If $f (1) = 2$ and $f (4) = 8$ then $f (2)$ has the value equal to