Rolle's Theorem The function f(x)={ 0, x=0 1-x, 0 < x ≤ 1 . is differentiable on (0,

step 1 In this problem, given a function of we know that f is equal to 0, 1x is equal to 0 and 1x is be

Rolle's Theorem The function f(x)={ 0, x=0 1-x, ...

Rolle's Theorem The function
$f(x)=\left\{\begin{array}{ll}{0,} & {x=0} \\ {1-x,} & {0 < x \leq 1}\end{array}\right.$
is differentiable on $(0,1)$ and satisfies $f(0)=f(1)$ . However, its derivative is never zero on $(0,1) .$ Does this contradict Rolle's Theorem? Explain.

Key Concepts

Endpoint Conditions

The condition that the endpoints of the function have equal values (f(a) = f(b)) is crucial for Rolle's Theorem. However, even if this condition is satisfied, the theorem will not hold if the function does not meet the continuity requirement at the endpoints. In cases where the function is discontinuous at an endpoint, the theorem's conclusion about the existence of a zero derivative point does not apply.

Differentiability on an Open Interval

Rolle's Theorem requires the function to be differentiable on the open interval (a, b), which ensures that there are no sharp corners or cusps within the interval. While differentiability on (a, b) is necessary, it alone is not enough; the function also needs to be continuous throughout the closed interval [a, b].

Rolle's Theorem

Rolle's Theorem is a result in calculus that states if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and has equal values at the endpoints (f(a) = f(b)), then there exists at least one point c in (a, b) where the derivative f?'(c) is zero. This theorem provides a fundamental link between the behavior of a function and its derivative on an interval.

Continuity on a Closed Interval

One of the key hypotheses of Rolle's Theorem is that the function must be continuous on the closed interval [a, b]. This means the function should not have any jump discontinuities or other forms of discontinuity at any point within the interval, including at the endpoints. Failing this condition means the theorem cannot be applied.

Transcript

Please give Ace some feedback