1. Tardigrade
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  3. Mathematics
  4. 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

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

Application of Derivatives

Solution:

ByL.M.V.T.
$f ^{\prime}( c )=\frac{ f (4)- f (2)}{4-2}=\frac{ f (4)+4}{2} $
$\text { But } f ^{\prime}( x ) \geq 6 \forall x \in[2,4] $
$\therefore f ^{\prime}( c ) \geq 6 \Rightarrow \frac{ f (4)}{2}+2 \geq 6 \Rightarrow f (4) \geq 8 .$