Let f be a function which is continuous and differentiable for all real x. If f (2)=-4 and f prime( x ) ≥ 6 for all x ∈[2,4], then

Question

Let f be a function which is continuous and differentiable for all real x. If f (2)=-4 and f prime( x ) ≥ 6 for all x ∈[2,4], then (A) f (4)<8 (B) f (4) ≥ 8 (C) f(4) ≥ 12 (D) f (4)<12

Tardigrade Admin · Accepted Answer

ByL.M.V.T. f prime( c )=( f (4)- f (2)/4-2)=( f (4)+4/2) text But f prime( x ) ≥ 6 ∀ x ∈[2,4] ∴ f prime( c ) ≥ 6 ⇒ ( f (4)/2)+2 ≥ 6 ⇒ f (4) ≥ 8 .

Q. Let $f$ be a function which is continuous and differentiable for all real $x$ . If $f (2) = - 4$ and $f^{'} (x) \geq 6$ for all $x \in [2, 4]$ , then

f $(4) < 8$

$f (4) \geq 8$

$f (4) \geq 12$

f $(4) < 12$

ByL.M.V.T.
$f^{'} (c) = \frac{f ( 4 ) - f ( 2 )}{4 - 2} = \frac{f ( 4 ) + 4}{2}$
$But f^{'} (x) \geq 6\forall x \in [2, 4]$
$∴ f^{'} (c) \geq 6 \Rightarrow \frac{f ( 4 )}{2} + 2 \geq 6 \Rightarrow f (4) \geq 8.$

Q. Let f be a function which is continuous and differentiable for all real x. If f(2)=−4 and f′(x)≥6 for all x∈[2,4], then

f (4)<8

f(4)≥8

f(4)≥12

f (4)<12