0Mean value theorem in simple terms:
1. If a real valued function f(x) is continuous over an closed interval [a,b]
2. And this function is differentiable over an open interval (a,b)
3. f(a) ≠ f(b) : ( if f(a) = f(b) then f'(x) =0 means function satisfies Rolle's theorem )!!!
4. Then there exists atleast one number 'c' between "a" and "b"( a < c < b ) which satisfies the equation f'(x)= [ f(a) - f(b)/b - a].
A function is said to be continuous if we can draw the curve without lifting the pen.
A function is said to be differentiable over an interval (a,b) if we can obtain the slope of the tangent to the curve at each and every point or every value of 'x'. For example take a function f(x) = |x|, if we plot a graph of this function it looks like alphabet " V " and now as we observe, we can obtain the slope at all the points except at the point where both limbs meet !!! So it is differentiable at all points except at where both limbs meet !!!!
f(a) ≠ f(b) means value of 'y' at 'x' = a is not equal to value of 'y' at 'x'= b.
One final clarification, closed interval between " a & b " means all the points between " a & b " including points " a & b ". Open interval between " a & b " means all the points between " a & b" but points " a & b" are not included.
A simple example just to understand the theory : let us assume a function when plotted resembles a slightly tilted semicircle on the X-Y plane and we join the end points which becomes the secant line, theorem states that when all the conditions mentioned above are fulfilled, then we will get atleast one point 'c' on this curve through which if a tangent is drawn it will be parallel to the secant line !!!!! # means slope of the tangent line at point 'c' is equal to the slope of the secant line #
