Question
In: Advanced Math
Rolle's Theorem Question if f(a)=f(b) then it is constant function, so every point c in the...
Rolle's Theorem Question
if f(a)=f(b) then it is constant function, so every point c in the interval (a,b) is zero.
I don't understand the terms constant function, is it supposed to be a horizontal line or it can be a parabola curve.
For example, when f(a)=f(b) and they both exist at the endpoint then it can be a parabola curve then it break the assumption that f(a)=f(b) then it is constant function, so every point c in the interval (a,b) is zero.
Please draw the picture to describe and explain
Solutions
Colby Messinger answered 2 years agoRelated Solutions
Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...
Rolle's Theorem, "Let f be a continuous function on [a,b] that
is differentiable on (a,b) and such that f(a)=f(b). Then there
exists at least one point c on (a,b) such that f'(c)=0."
Rolle's Theorem requires three conditions be satisified.
(a) What are these three conditions?
(b) Find three functions that satisfy exactly two of these three
conditions, but for which the conclusion of Rolle's theorem does
not follow, i.e., there is no point c in (a,b) such that f'(c)=0.
Each...
Determine whether Rolle's Theorem applies to the following function on the given interval. If so, find...
Determine whether Rolle's Theorem applies to the following
function on the given interval. If so, find the point(s) that are
guaranteed to exist by Rolle's Theorem. g(x)=x^3- 9 x^2 + 24 x -
20; [2,5]
A)Rolle's Theorem applies and the point(s) guaranteed to exist
is/arex=
(Type an exact answer, using radicals as needed. Use a comma
to separate answers as needed.)
B)Rolle's Theorem does not apply.
1. Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]....
1.
Determine whether Rolle's Theorem can be applied to
f
on the closed interval
[a, b].
(Select all that apply.)
f(x)
=
x2− 4x
− 5
x + 5
, [−1,
5]
Yes,
Rolle's Theorem can be applied.No, because f is
not continuous on the closed interval [a,
b].No, because f is
not differentiable in the open interval (a,
b).No, because
f(a) ≠
f(b).
If Rolle's Theorem can be applied, find all values of
c
in the open interval
(a, b)...
2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for...
2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all
xin an interval ( a , b ), then f is constant on ( a , b
).
b True or False. The product of two increasing functions is
increasing. Clarify your answer.
c Find the point on the graph of f ( x ) = 4 − x 2 that is
closest to the point ( 0 , 1 ).
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then...
Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers
c that satisfy the conclusion of Rolle's Theorem. (Enter
your answers as a comma-separated list.)
f(x) = cos(2x), [π/8, 7π/8]
(a) Graph the function
f(x) = x +
7/x
and the secant line that passes through the points (1, 8) and
(14, 14.5) in the viewing rectangle [0, 16] by [0, 16].
(b) Find the number c that satisfies the conclusion of...
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then...
Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers
c that satisfy the conclusion of Rolle's Theorem. (Enter
your answers as a comma-separated list.)
f(x) = 1 − 24x +
4x2 ,[2,4]
C=
1) A) Determine whether Rolle's Theorem can be applied to f on the closed interval [a,...
1)
A) Determine whether Rolle's Theorem can be
applied to f on the closed interval [a, b]. (Select all
that apply.)
f(x) = (x − 1)(x − 2)(x − 8), [1, 8]
Yes, Rolle's Theorem can be applied.
No, because f is not continuous on the closed interval
[a, b].
No, because f is not differentiable in the open
interval (a, b).
No, because f(a) ≠ f(b).
B) If Rolle's Theorem can be applied, find all
values of c in the...
Does the given function satisfy the three hypotheses of Rolle's Theorem on the given interval? If...
Does the given function satisfy the three hypotheses of Rolle's
Theorem on the given interval? If so, determine where f ′ ( x ) = 0
within the interval; if not, explain which hypotheses are not
satisfied.
According to the Fundamental Theorem of Algebra, every nonconstant polynomial f (x) ∈ C[x] with complex...
According to the Fundamental Theorem of Algebra, every
nonconstant polynomial f (x) ∈
C[x] with complex coefficients has a complex root.
(a) Prove every nonconstant polynomial with complex coefficients
is a product of linear polynomials.
(b) Use the result of the previous exercise to prove every
nonconstant polynomial with real coefficients is a product of
linear and quadratic polynomials with real coefficients.
Prove by induction that it follows from Fundamental Theorem of Algebra that every f(x) ∈ C[x]
Prove by induction that it follows from Fundamental Theorem of Algebra that every f(x) ∈ C[x] can be written into a product of linear polynomials in C[x].
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