If $f$ and $g$ are differentiable functions in $(0, 1)$ satisfying
$f (0) = 2 = g (1), g (0) = 0$ and $f (1) = 6$, then for some
$c \in ] 0, 1 [$
- $2f ' (c) = g ' (c)$
- $2f ' (c) = 3 g ' (c)$
- $f ' (c) = g ' (c)$
- $f ' (c) = 2 g ' (c)$
The Correct Option is D
Solution and Explanation
$f'(c)=\frac{f(1)-f(0)}{1-0}=4$
$g'(c)=\frac{g(1)-g(0)}{1-0}=2$
so, $f'(c)=2 g'(c)$
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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.