gruba said:Homework Statement
Let [itex]f[/itex] is differentiable function on [itex][0,1][/itex] and [itex]f^{'}(0)=1,f^{'}(1)=0[/itex]. Prove that [itex]\exists c\in(0,1) : f^{'}(c)=f(c)[/itex].
Homework Equations
-Mean Value Theorem
The Attempt at a Solution
The given statement is not true. Counter-example is [itex]f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10[/itex].
Does this mean that the statement can't be proved?
The Mean Value Theorem Variation Proof is a mathematical concept that states that if a function is continuous on an interval and differentiable on the open interval, then there exists at least one point within the interval where the slope of the tangent line is equal to the average rate of change of the function.
The Mean Value Theorem Variation Proof is used to prove that a function has a specific property, such as a maximum or minimum value, within a given interval. It is also used to prove other theorems in calculus, such as the Fundamental Theorem of Calculus.
The Mean Value Theorem Variation Proof requires that the function is continuous on the interval and differentiable on the open interval. Additionally, the function must also have a starting and ending point within the interval.
No, the Mean Value Theorem Variation Proof can only be used for functions that meet the necessary criteria of being continuous and differentiable on the given interval. Functions that are not continuous or differentiable may not have a point where the slope of the tangent line is equal to the average rate of change.
The Mean Value Theorem Variation Proof is a generalization of Rolle's Theorem. Both theorems state that if a function meets certain criteria, then there exists at least one point within the interval where the slope of the tangent line is equal to zero. However, the Mean Value Theorem Variation Proof allows for the possibility of the slope being equal to any given value, while Rolle's Theorem only states that the slope must be equal to zero.