Mean Value Theorem for Continuous Functions
Key Questions

The value of
#c# is#sqrt{3}# .Let us look at some details.
M.V.Thm. states that there exists
#c# in (0,3) such that#f'(c)={f(3)f(0)}/{30}# .Let us find such
#c# .The lefthand side is
#f'(c)=3c^2+1# .The righthand side is
#{f(3)f(0)}/{30}={29(1)}/{3}=10# .By setting them equal to each other,
#3c^2+1=10 Rightarrow 3x^2=9 Rightarrow x^2=3 Rightarrow x=pm sqrt{3}# Since
#0<c<3# ,#c=sqrt{3}# .I hope that this was helpful.

Actually, Rolle's Theorem require differentiablity, and it is a special case of Mean Value Theorem.
Please watch this video for more details.

Mean Value Theorem
If a function#f# is continuous on#[a,b]# and differentiable on#(a,b)# ,
then there exists c in#(a,b)# such that#f'(c)={f(b)f(a)}/{ba}# .
Questions
Graphing with the First Derivative

Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Identifying Stationary Points (Critical Points) for a Function

Identifying Turning Points (Local Extrema) for a Function

Classifying Critical Points and Extreme Values for a Function

Mean Value Theorem for Continuous Functions