%%visits: 2 ## intuition mean_value_theorem application to area analysis
f : (a, b) → ℝ is differentiable. If an extremum is reached, then there exists a point in the interval, c such that f′(c) = 0
Sketch, left and right limits exists, and left and right are opposite signs, so must be 0.
non-example of extreme_value_theorem.
f(x) = −|x| ≤ 0. Extremised (extremum) at x=0 but not differentiable.
mean_value_theore := Let f ∈ C[a, b]
be differentiable on (a, b). Then there exists
c ∈ (a, b)
such that $f(c) = \frac{f(b) -
f(a)}{b-a}$
mean_value_theorem:~ The line that bisects two points on a smooth graph, has the same gradient as at least 1 point on that line.
proof of mean_value_theorem. Let $F(t) = (f(t) - f(a)) - ( ) (t-a) $ (this is a linear approximate) as t → a then it tends to 0 :todo: ## rigour ## exam clinic ## examples and non-examples ## resources tags :math: