Question

In: Advanced Math

NOTE- If it is true, you need to prove it and If it is false, give...

NOTE- If it is true, you need to prove it and If it is false, give a counterexample

f : [a, b] → R is continuous and in the open interval (a,b) differentiable.

a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0.(TRUE or FALSE?)

b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?)
c) If f ′ is limited, then f is lipschitz. (TRUE or FALSE?)

Solutions

Expert Solution

a) The statement is true

If the function is a constant, it has zero slope everywhere and then we are done.
If it is not a constant it is decreasing somewhere at which the slope is negative. So in both of these cases the statement is true. Hence the statement is true.
b) The statement is true

Consider an invertible function f pocessing the derivative f'. Since the function is invertible it is one-one and onto.
So we have,

Diffrentiating both sides with respect to x ,

By chain rule.
So,

So the condition is that

Any function violating this condition would raise a problem. But our function is one-one so it can not have zero slope anywhere. ( If it pocess zero slope somewhere there would be many points in [a,b] mapping to a single point in the range). So the derivative of the inverse function exists.

c) The statement is true

Let f' be limited. i.e, let for some positive constant C.

let x and y belong to [a,b]. Now since f is diffrentiable, by mean value theorem , there exist a c in (x,y) so that

Now by taking modulus on both sides and by replacing by the heavier , we get the lipschitz condition,


Related Solutions

NOTE- If it is true, you need to prove it and If it is false, give...
NOTE- If it is true, you need to prove it and If it is false, give a counterexample f : [a, b] → R is continuous and in the open interval (a,b) differentiable. a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0. (TRUE or FALSE?) b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?) c) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)
Determine whether each statement is true or false. If it is true, prove it. If it...
Determine whether each statement is true or false. If it is true, prove it. If it is false, give a counterexample. a) For every function f : X → Y and all A ⊆ X, we have f^−1 [f[A]] = A. (b) For every function f : X → Y and all A ⊆ X, we have f[X \ A] = Y \ f[A]. (c) For every function f : X → Y and all A, B ⊆ Y ,...
(need in frame -c) In the given code, Will all the assert statements prove true? Give...
(need in frame -c) In the given code, Will all the assert statements prove true? Give reason for proved true or false for each assert statement. [3 M] [CO2] int main(){ int i = 1; int x = 33; /*@ loop invariant 1 <= i <= 19; loop assigns i; loop variant 20 - i; */ //@assert i == 1; while(i < 20){ ++i; if(i == 20) break ; } //@assert i == 20; //@assert x == 33; } frama-c...
1. Determine if the following statements are true or false. If a statement is true, prove...
1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample. Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation. A) If φ is injective, then dim(V) ≤ dim(W). B) If dim(V) ≤ dim(W), then φ is injective. C) If φ is surjective, then dim(V) ≥ dim(W). D) If dim(V) ≥...
The following are True or False statements. If True, give a simple justification. If False, justify,...
The following are True or False statements. If True, give a simple justification. If False, justify, or better, give a counterexample. 1. (R,discrete) is a complete metric space. 2. (Q,discrete) is a compact metric space. 3. Every continuous function from R to R maps an interval to an interval. 4. The set {(x,y,z) : x2 −y3 + sin(xy) < 2} is open in R3
Indicate whether the statement is true or false. If false, please explain why. (Note that if...
Indicate whether the statement is true or false. If false, please explain why. (Note that if the explanation is incomplete, you may not get the full credits.) a. Although payroll taxes are imposed on firms, workers bear more burdens than firms when the elasticity of labor supply is greater than the elasticity of labor demand. (4 pts) b. There might be a situation in which dangerous jobs offer lower wages than safe jobs even if there exist compensating wage differentials....
This is true and false and if the asnwer is false i need the correct answer....
This is true and false and if the asnwer is false i need the correct answer. _____ The least squares slope for predicting y from x is the same asthe least squares slope for predicting xfrom y _____ The least squares linepasses through the point of averages . _____ A residual is the deviation of a point from the point of averages _____ It is common to use a residual plotbased on plotting the residuals versus x. ______Theleast squares lineis...
True or False. Please give an explanation as to why its true or false. Too much...
True or False. Please give an explanation as to why its true or false. Too much leverage on a company’s balance sheet will increase the company’s cost of equity/
Write a formal proof to prove the following conjecture to be true or false. If the...
Write a formal proof to prove the following conjecture to be true or false. If the statement is true, write a formal proof of it. If the statement is false, provide a counterexample and a slightly modified statement that is true and write a formal proof of your new statement. Conjecture: Let w, x, y, and z be single-digit numbers. The 4-digit number wxyz* is divisible by 9 if and only if 9 divides the sum w + x +...
Determine whether the statement is true or false. If it is false, explain why or give...
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(c) = L,then lim x→c f(x) = L. False. Define f to be the piece-wise function where f(x) = x + 3 when x ≠ −1 and f(x) = 2 when x = −1. Then we have f(−1) = 2 while the limit of f as x approaches −1 is equal to −2. False. Define f...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT