Question:
Let $f ' (x),$ be differentiable $\forall \, x.$ If $f (1) = -2$ and $f '(x) \geq 2 \forall x \in [1, 6],$ then
Let $f ' (x),$ be differentiable $\forall \, x.$ If $f (1) = -2$ and $f '(x) \geq 2 \forall x \in [1, 6],$ then
Updated On: Apr 15, 2024
- $f (6) < 8 $
- $f (6) \geq 8$
- $f (6) \geq 5 $
- $f (6) \leq 8$
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The Correct Option is B
Solution and Explanation
Since, $f' (x) $ is differentiable $\forall x \in [1, 6]$
$\therefore$ By Lagrange?s mean value theorem,
$f'(x) = \frac{f(6) - f(1)}{6 - 1}$
$\because f'(x) \geq 2 \forall x \in [1, 6 ]$ (given )
$\Rightarrow \,\therefore \frac{f(6) + 2 }{5} \geq 2$ [$\because \, f(1) = - 2$]
$\Rightarrow f(6) \geq 10 - 2 \, \Rightarrow f(6) \geq 8 $
$\therefore$ By Lagrange?s mean value theorem,
$f'(x) = \frac{f(6) - f(1)}{6 - 1}$
$\because f'(x) \geq 2 \forall x \in [1, 6 ]$ (given )
$\Rightarrow \,\therefore \frac{f(6) + 2 }{5} \geq 2$ [$\because \, f(1) = - 2$]
$\Rightarrow f(6) \geq 10 - 2 \, \Rightarrow f(6) \geq 8 $
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.