The function f(x)={ x, 0 ≤ x<1 0, x=1 . is zero at x=0 and x=1 and differentiable on

step 1 In this problem we're given a function where f is equal to x when x is between 0 and 1 1 and whe

The function f(x)={ x, 0 ≤ x<1 0, x=1 . is zero...

The function $$f(x)=\left\{\begin{array}{ll}{x,} & {0 \leq x<1} \\ {0,} & {x=1}\end{array}\right.$$ is zero at $x=0$ and $x=1$ and differentiable on $(0,1),$ but its derivative on $(0,1)$ is never zero. How can this be? Doesn't Rolle's Theorem say the derivative has to be zero somewhere in $(0,1) ?$ Give reasons for your answer.

The function
$$f(x)=\left\{\begin{array}{ll}{x,} & {0 \leq x<1} \\ {0,} & {x=1}\end{array}\right.$$
is zero at $x=0$ and $x=1$ and differentiable on $(0,1),$ but its derivative on $(0,1)$ is never zero. How can this be? Doesn't Rolle's Theorem say the derivative has to be zero somewhere in $(0,1) ?$ Give reasons for your answer.

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Key Concepts

Rolle's Theorem

Rolle's Theorem is a special case of the Mean Value Theorem. It states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and takes equal values at the endpoints (i.e., f(a) = f(b)), then there is at least one point in (a, b) where the derivative is zero. This theorem emphasizes the necessity of all its conditions being met for its conclusion to hold.

Continuity on a Closed Interval

A key condition for applying Rolle's Theorem is that the function must be continuous on the closed interval [a, b]. This ensures that there are no jumps, breaks, or removable discontinuities within the interval. In the provided function example, the failure of continuity at one of the endpoints means the theorem is not applicable.

Differentiability on an Open Interval

Another important condition is that the function must be differentiable on the open interval (a, b). Differentiability implies the function has a well-defined slope everywhere within the interval. However, even if a function is differentiable on (a, b), without the continuity on [a, b] (including the endpoints), the conclusions of Rolle's Theorem do not necessarily follow.

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