3. Use the mean value theorem to prove the following inequality. (1 +x)" >1 for z >0 andnEN
3. Use the mean value theorem to prove the following inequality. (1 +x)" >1 for z >0 andnEN
5. Use the mean value theorem to prove that cos x - cosyl < x - y for x,y E R.
1. (a) State and prove the Mean-Value Theorem. You may use Rolle's Theorem provided you state it clearly (b) A fired point of a function g: (a, bR is a point cE (a, b) such that g(c)-c Suppose g (a, b is differentiable and g'(x)< 1 for all x E (a, b Prove that g cannot have more than one fixed point. <「 for (c) Prove, for all 0 < x < 2π, that sin(x) < x.
let a > 0 and define g(x) := x^(a+1) - (a+1)x + a. Use the mean value theorem to show that g(x)>0 for all x>0, where x~=1 3. Let α > 0 and define g(x):-Χα +1-(α + 1)x + α. Use the mean value theorem to show that (x>for allx >0, where x I 3. Let α > 0 and define g(x):-Χα +1-(α + 1)x + α. Use the mean value theorem to show that (x>for allx >0, where x...
Use the Mean Value Theorem to demonstrate that In(1 + x) < x, given that x > 0.
Use perfect induction to prove Theorem 7:( x + y ) ( x ′ + z ) = x z + x ′ y .
Analysis 6. (a) State the Mean Value Theorem. Calo o- (b) Use your answer to (a) to prove that if f(x) is differentiable on [0,3), f(0)-5, and f(x) >3 for all z (0,3), then ()> 5+3r for all e [o,3].
Real analysis subject 6. Prove the following slight generalization of the Mean Value Theorem: if f is continuous and differentiable on (a, b) and limy a f(v) and limyb- f(s) exist, then there is some z in (a, b) such that -a (Your proof might begin: "This is a trivial consequence of the Mean Value Theorem because ...".) .. 6. Prove the following slight generalization of the Mean Value Theorem: if f is continuous and differentiable on (a, b) and...
Use Jensen's inequality to prove that the arithmetic mean is at least as larg e as the geometric mean. That is, for nonnegative x, show that 1in
Use the Mean Value Theorem to demonstrate there is at least one root for f(x)=x^3+x-1 on [0,2]. Find the area between the curve x^3-3x+ 3 and the x‐axis on the interval [1, 3].