Proving e^x ≥ 1 + x for All x ∈ [0,∞)

  • Context: Undergrad 
  • Thread starter Thread starter nuuc
  • Start date Start date
  • Tags Tags
    E^x
Click For Summary
SUMMARY

The inequality e^x ≥ 1 + x holds true for all x in the interval [0, ∞). To prove this, define the function f(x) = e^x - x - 1. By calculating the first derivative, f'(x) = e^x - 1, we can analyze the behavior of f(x). The first derivative test indicates that f'(x) is non-negative for x ≥ 0, confirming that f(x) is always greater than or equal to zero in this interval.

PREREQUISITES
  • Understanding of calculus, specifically derivatives
  • Familiarity with exponential functions
  • Knowledge of the first derivative test
  • Basic concepts of function behavior and intervals
NEXT STEPS
  • Study the properties of exponential functions and their derivatives
  • Learn about the first derivative test in detail
  • Explore the concept of limits and continuity in calculus
  • Investigate other inequalities involving exponential functions
USEFUL FOR

Students of calculus, mathematics educators, and anyone interested in understanding inequalities involving exponential functions.

nuuc
Messages
2
Reaction score
0
how would i prove that e^x >= 1 + x for all x in [0,inf)?
 
Physics news on Phys.org
where are your thoughts?
hint: expand out to see.
 
Define f(x) = e^x - x - 1. What can f'(x) tell us?
 
morphism said:
Define f(x) = e^x - x - 1. What can f'(x) tell us?
like morphism stated, now what u have to do is just find f'(x)= (e^x-x-1)', what is f'(x) now?,
After that look at what interval the function f(x) is always greater or equal to zero, using the first derivative test,can you do it? and the result surely should be for any x greater than zero. It is obvious that for x=0 the >= sing is valid.
 

Similar threads

Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
911
High School Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Significance of (1+1/x)^x as x->-inf
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
MHB Is the Limit of Sin x/x=1 Proven in Elementary Calculus?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Solving the Difficult Integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##
  • · Replies 5 ·
Replies
5
Views
3K
MHB How can i integrate this function
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solve Complex Integration: Find 2.36 Area of y=-x/2e+1/e+e & y=e^x/2
  • · Replies 10 ·
Replies
10
Views
2K