How do you use the Mean Value Theorem to prove Bernoulli's inequality?
which completes the proof.
By signing up, you agree to our Terms of Service and Privacy Policy
To prove Bernoulli's inequality using the Mean Value Theorem, follow these steps:

Statement of Bernoulli's Inequality: Bernoulli's inequality states that for any real number ( x ) greater than or equal to 1 and for any natural number ( n ), [ (1 + x)^n \geq 1 + nx ]

Use of Mean Value Theorem: Consider the function ( f(t) = (1 + t)^n ) for ( t ) in the interval [0, x]. Apply the Mean Value Theorem, which states that if a function ( f ) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number ( c ) in (a, b) such that: [ f'(c) = \frac{f(b)  f(a)}{b  a} ]

Application to Bernoulli's Inequality: For our function ( f(t) = (1 + t)^n ) and interval [0, x], we have: [ f'(c) = n(1 + c)^{n1} ] Applying the Mean Value Theorem: [ f'(c) = \frac{f(x)  f(0)}{x  0} ] Substituting our function: [ n(1 + c)^{n1} = \frac{(1 + x)^n  1}{x} ]

Rearrange to Prove Bernoulli's Inequality: Multiply both sides by ( x ): [ nx(1 + c)^{n1} = (1 + x)^n  1 ] As ( c ) is between 0 and ( x ), ( 1 + c ) is between 1 and ( 1 + x ). Hence, ( (1 + c)^{n1} \geq 1 ). Thus, ( nx(1 + c)^{n1} \geq nx ). This implies: [ (1 + x)^n  1 \geq nx ] Adding 1 to both sides: [ (1 + x)^n \geq 1 + nx ]
This completes the proof of Bernoulli's inequality using the Mean Value Theorem.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 What are the global and local extrema of #f(x)=x^2 2x +3# ?
 What are the absolute extrema of #f(x)=(9x^(1/3))/(3x^21) in[2,9]#?
 What are the values and types of the critical points, if any, of #f(x,y)=4y(1  x^2)#?
 How do you find the local extremas for #g(x) =  x+6#?
 What are the global and local extrema of #f(x)=8x^34x^2+6# ?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7