Notes On Mean Value Theorem - CBSE Class 12 Maths
The graphs of a parabola and a sine function.  f(x) and g(x) are continuous in [a,b]. The parabola has only one tangent parallel to the X-axis and the sine graph has two tangents parallel to the X-axis. The slopes of tangents parallel to the X-axis are zero. In each of the graphs, the slope becomes zero at one point at least. Rolle's Theorem: If f: [a,b] → R is continuous on [a,b] and differentiable on (a,b), such that f(a) = f(b), where a and b are some real numbers, then there exists some c in (a,b), such that f '(c) = 0. This theorem claims that the slope of a tangent at any point on the graph of y = f(x) is the derivative of f(x) at that point. If f(x) be a continuous function on [a,b] and differentiable on (a,b), where f(a) ≠ f(b) then the Rolle's theorem is not satisfied. Mean Value Theorem [MVT] : If f:[a,b] → R is continuous on [a,b] and differentiable on (a,b), then there exists some c in (a,b), such that f '(c) = (f(b) - f(a)) / (b-a). f '(c) is the slope of the tangent to y = f(x) at (c,f(c)). Slope of the secant = (f(b) - f(a)) / (b-a) The MVT states that there exists some c ∈ (a,b), such that the tangent at (c,f(c)) is parallel to the secant that joins (a,f(a)) and (b,f(b)).

#### Summary

The graphs of a parabola and a sine function.  f(x) and g(x) are continuous in [a,b]. The parabola has only one tangent parallel to the X-axis and the sine graph has two tangents parallel to the X-axis. The slopes of tangents parallel to the X-axis are zero. In each of the graphs, the slope becomes zero at one point at least. Rolle's Theorem: If f: [a,b] → R is continuous on [a,b] and differentiable on (a,b), such that f(a) = f(b), where a and b are some real numbers, then there exists some c in (a,b), such that f '(c) = 0. This theorem claims that the slope of a tangent at any point on the graph of y = f(x) is the derivative of f(x) at that point. If f(x) be a continuous function on [a,b] and differentiable on (a,b), where f(a) ≠ f(b) then the Rolle's theorem is not satisfied. Mean Value Theorem [MVT] : If f:[a,b] → R is continuous on [a,b] and differentiable on (a,b), then there exists some c in (a,b), such that f '(c) = (f(b) - f(a)) / (b-a). f '(c) is the slope of the tangent to y = f(x) at (c,f(c)). Slope of the secant = (f(b) - f(a)) / (b-a) The MVT states that there exists some c ∈ (a,b), such that the tangent at (c,f(c)) is parallel to the secant that joins (a,f(a)) and (b,f(b)).

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