MEAN VALUE AND ROLLE'S THEOREM

MEAN VALUE AND ROLLE'S THEOREM

1. ROLLE'S THEOREM

   • Let f be a real valued function defined on the closed interval [a, b] such that
     • GEOMETRICAL INTERPRETATION OF ROLLE'S THEOREM
       • Let f(x) be a real valued function defined on [a, b] such that the curve y = f(x) is a continuous curve between points A(a, f (a)) and B(b, f(b)) and it is possible to draw a unique tangent at every point on the curve y = f(x) between points A and B. Also, the ordinates at the end points of the interval [a, hi are equal. Then, there exists at least one point (c, f(c)) lying between A and B on the curve y = f(x) where tangent is parallel to x-axis. Y (c, f(c)) A(a, f(a))
         • GRAPH
     • ALGEBRAIC INTERPRETATION OF ROLLE'S THEOREM
       • Thus, algebraically Rolle's theorem can be interpreted as follows: Between any two roots of a polynomial f (x), there is always a root of its derivative f ' (x). REMARK
         • On Rolle's theorem generally two types of problems are formulated.
           • (a) To check the applicability of Rolle's theorem to a given function on a given interval,
           • (b) To verify Rolle's theorem for a given function on a given interval.
             • In both types of problems we first check whether f(x) satisfies conditions of Rolle's theorem or not.
               • A polynomial function is everywhere continuous and differentiable.
               • The exponential function, sine and cosine functions are every where continuous and differentiable.
               • Logarithmic function is continuous and differentiable in its domain.
               • tan x is not continuous at x = ± 2, ± 3n/2, ± 5n/2•
               • |x| is not differentiable at x=0
               • The sum ,difference ,product ,quotient of continuous function is continuous(differentiable)
               • if f '(x) tends to +- ∞ as x tends to k, then f(x) is not fifferentiable at x=k
     • (i) it is continuous on the closed interval [a, b]
       • , (ii) it is differentiable on the open interval (a, b), and, (iii)
         • f(a) = f(b)•
           • Then, there exists a real number c E (a, b) such that f ' (c) = 0.

2. MEAN VALUE THEOREM

   • Let f(x) be a function defined on [a,b] such that
     • (i) it is continuous on [a,b]
       • (ii) it is differentiable on (a,b)
         • Then there exist a real number c E(a,b) such that f '(c) = f(b)-f(a)/(b-a) therefore, tanɸ=f "(c)which implies slope of the chord =slope of tangent
   • GEOMETRICAL INTERPRETATION
     • Let f(x) be a real function defined on [a, b] such that the curve y = f(x) is a continuous curve between points A(a, f (a)) and B(b, f(b)) at every point on the curve ,except the end points ,except at the end points ,it is possible to draw a unique tangent .Then there exists a point on the curve such that the tangent is parallel to the chord joining the end points of the curve.

3. CONVERSE OF ROLLE'S THEOREM (NOT APPLICABLE)

   • If f[a,b]--R for some c E [a,b] for which f'c)=0 then
     • (i)f(a)=f(b)
       • (ii) f is continuous at [a,b]
         • (iii)and differentiable at [a,b]

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