Mean Value Theorems — Formula Sheet

f'(x) = slope of the tangent to y = f(x) at (x, f(x))

f'(x) = 0

Slope of secant AB = (f(b) - f(a)) / (b - a)

f'(c) = 0, for some c in (a,b)

f'(c) = (f(b) - f(a)) / (b - a), for some c in (a,b)

f(x)=tan x on [0,π] is not applicable because the function is not continuous at x=π/2.

f(x)=√(5-x) on [-3,6] is not applicable because the function is not continuous on the entire interval.

f(x)=⌊x⌋ on [5,9] is not applicable because it is discontinuous at integer points such as x=6.

For f(x)=x^3+ax^2+bx on [1,2] with c=4/3, the values are a=-5 and b=8.

There can be more than one point where f'(c) = 0.

If f:[a,b]→R is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c∈(a,b) such that f'(c)=0.

If f:[a,b]→R is continuous on [a,b] and differentiable on (a,b), then there exists c∈(a,b) such that f'(c)=(f(b)-f(a))/(b-a).