Solving Simple Inequality: Tips/Suggestions Needed

  • Thread starter autobot.d
  • Start date
  • Tags
    Inequality
In summary, the conversation discusses a method for proving the inequality \left(a+b\right)^{p} \leq a^{p}+b^{p} for 0 < p < 1 and a,b\geq 0 without using differentiation. The method involves using the binomial theorem, but it is pointed out that p is not an integer. It is then suggested to use Newton's generalization, which is explained further on Wikipedia.
  • #1
autobot.d
68
0
Is there a way to do this without differentiation?

[itex]\left(a+b\right)^{p}[/itex] [itex]\leq a^{p}+b^{p}[/itex]


0<p<1 and a,b[itex]\geq[/itex] 0

pulling the a out of the the first part and dividing by it to get

[itex]\left(1+\frac{b}{a}\right)^{p}[/itex][itex]\leq 1+\frac{b}{a}^{p}[/itex]

This seems like the way to go but am stuck. Any suggestions? Thanks.
 
Physics news on Phys.org
  • #2
autobot.d said:
Is there a way to do this without differentiation?

[itex]\left(a+b\right)^{p}[/itex] [itex]\leq a^{p}+b^{p}[/itex]


0<p<1 and a,b[itex]\geq[/itex] 0

pulling the a out of the the first part and dividing by it to get

[itex]\left(1+\frac{b}{a}\right)^{p}[/itex][itex]\leq 1+\frac{b}{a}^{p}[/itex]

This seems like the way to go but am stuck. Any suggestions? Thanks.

Use the binomial theorem?
 
  • #3
p is not an integer though. Not sure binomial thm would work.

0 < p < 1
 
  • #4
autobot.d said:
p is not an integer though. Not sure binomial thm would work.

0 < p < 1

It works.
 
  • #5
Did not know that, will do some research.
 
  • #6
autobot.d said:
Did not know that, will do some research.

It is Newton's generalisation that works. It is there on wikipedia.
 

1. What is an inequality?

An inequality is a mathematical statement that compares two quantities using symbols such as <, >, ≤, ≥, or ≠. It shows the relationship between the two quantities and indicates which one is larger or smaller.

2. How do I solve a simple inequality?

To solve a simple inequality, you need to isolate the variable on one side of the inequality sign. You can do this by using inverse operations, which means performing the opposite operation on both sides of the inequality. Remember to follow the rules of inequality, such as flipping the sign when multiplying or dividing by a negative number.

3. What are some tips for solving inequalities?

Some tips for solving inequalities include identifying the variable, using inverse operations, and graphing the solution on a number line. It is also important to pay attention to any rules of inequality, such as flipping the sign when multiplying or dividing by a negative number, and to always check your solution by plugging it back into the original inequality.

4. Can you give an example of solving a simple inequality?

Sure, let's solve the inequality 2x + 5 < 15. First, we isolate the variable by subtracting 5 from both sides, which gives us 2x < 10. Then, we divide both sides by 2 to get the solution of x < 5. To check our solution, we plug in x = 5 into the original inequality and see if it is true: 2(5) + 5 < 15, which simplifies to 15 < 15. Since this is not true, we know our solution of x < 5 is correct.

5. What are some common mistakes to avoid when solving inequalities?

Some common mistakes to avoid when solving inequalities include forgetting to flip the sign when multiplying or dividing by a negative number, not checking your solution, and not following the correct order of operations. It is also important to remember to keep the inequality sign pointing in the correct direction throughout the solving process.

Similar threads

I Express the limit as a definite integral
    • Calculus
Replies
16
Views
2K
I Finding a bound for a fraction
    • Calculus
Replies
24
Views
1K
B Proving the Existence and Value of a Derivative at a Point
    • Calculus
Replies
11
Views
967
B Let f_n denote an element in a sequence of functions that converges
    • Calculus
Replies
7
Views
1K
Insights A Novel Technique of Calculating Unit Hypercube Integrals
    • Calculus
Replies
1
Views
2K
I Computing the expectation of the minimum difference between the 0th i.i.d.r.v. and ith i.i.d.r.v.s where 1 ≤ i ≤ n
    • Set Theory, Logic, Probability, Statistics
Replies
0
Views
325
I Finding a limit involving Chebyshev polynomials
    • Calculus
Replies
4
Views
935
B Riemann integrability of functions with countably infinitely many dis-
    • Calculus
Replies
3
Views
1K
Chernoff Bounds using importance sampling identity
    • Set Theory, Logic, Probability, Statistics
Replies
0
Views
1K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
    • Calculus
Replies
9
Views
814

Hot Threads

Recent Insights

Back
Top