Answer:
Well, they are basically just facts: some result that has been arrived at.
A Theorem is a major result
A Corollary is a theorem that follows on from another theorem
A Lemma is a small result (less important than a theorem)
Examples
Here is an example from Geometry:
Example: A Theorem and a Corollary
Theorem:
Angles on one side of a straight line always add to 180°.
angles add to 180 degrees
Corollary:
Following on from that theorem we find that where two lines intersect, the angles opposite each other (called Vertical Angles) are equal (a=c and b=d in the diagram).
vertically-opposite-abcd
Angle a = angle c
Angle b = angle d
Proof:
Angles a and b add to 180° because they are along a line:
a + b = 180°
a = 180° − b
Likewise for angles b and c
b + c = 180°
c = 180° − b
And since both a and c equal 180° − b, then
a = c
Step-by-step explanation:
pa brainliest
Answer:
Let f be a continuous function over the closed interval [a,b] and differentiable over the open interval (a,b) such that f(a)=f(b). There then exists at least one c∈(a,b) such that f′(c)=0.
Proof
Let k=f(a)=f(b). We consider three cases:
f(x)=k for all x∈(a,b).
There exists x∈(a,b) such that f(x)>k.
There exists x∈(a,b) such that f(x)<k.
Case 1: If f(x)=0 for all x∈(a,b), then f′(x)=0 for all x∈(a,b).
Case 2: Since f is a continuous function over the closed, bounded interval [a,b], by the extreme value theorem, it has an absolute maximum. Also, since there is a point x∈(a,b) such that f(x)>k, the absolute maximum is greater than k. Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point c∈(a,b). Because f has a maximum at an interior point c, and f is differentiable at c, by Fermat’s theorem, f′(c)=0.
Case 3: The case when there exists a point x∈(a,b) such that f(x)<k is analogous to case 2, with maximum replaced by minimum.
An important point about Rolle’s theorem is that the differentiability of the function f is critical. If f is not differentiable, even at a single point, the result may not hold. For example, the function f(x)=|x|−1 is continuous over [−1,1] and f(−1)=0=f(1), but f′(c)≠0 for any c∈(−1,1) as shown in the following figure.
Step-by-step explanation:
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