Let the function $ f(x) $ be continuous in the interval $ [a, b] $ and differentiable in $ (a, b) $ . Then there is at least one point $ c $ in $ (a, b) $ at which the tangent to the curve $ y = f (x) $ is parallel to
- $ x $ - axis
- $ y $ - axis
- the straight line
- the chord joining the points $ (a, f (a)) $ and $ (b, f (b)) $
The Correct Option is D
Solution and Explanation
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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.