Proving f=0 on [a,b] with Continuous Nonnegative f

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In summary, the problem states that for a continuous non-negative function f mapping from [a,b] to R, if the integral of f(x) from a to b is equal to 0, then f must be equal to 0 on the interval [a,b]. This can be proven by showing that if f is not equal to 0, then the integral of f will be greater than 0, which contradicts the given statement. Continuity of f can also be used to show that a positive area rectangle can be placed under f, supporting the conclusion that f must be equal to 0 on [a,b].
  • #1
CarmineCortez
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Homework Statement



Let f map [a,b]-->R be a continuous nonegative function. Suppose Integral f(x)dx from a to b = 0 show that f = 0 on [a,b]


The Attempt at a Solution



Just not sure if this is good or not..

so the lower sum <= 0 = integral f(x) dx

but the lower sum must be 0 since f is non negative.

now if f >0 then the lower sum >0 so this is a contradiction, f = 0.
 
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  • #2
you need to work the fact that f is continuous in there.

If you consider a function i.e f(x)= 1 for rationals , 0 for irrationals; the integral of this is zero, the lower sum is zero, but f is in no way 0 throughout the entire interval.
 
  • #3
Sort of. The idea is right. But you didn't use that f is continuous, and if f isn't continuous, it's not true. Show that if f is NOT equal to 0, then that implies the integral of f is greater than zero. Pick a point x where f(x)>0. Can you see how to use continuity at x to show you can put a positive area rectangle under f (like in a lower sum)?
 

What does it mean to prove f=0 on [a,b] with continuous nonnegative f?

Proving f=0 on [a,b] with continuous nonnegative f means to show that for any x in the interval [a,b], the function f(x) has a value of 0 and that the function remains continuous and nonnegative throughout the interval.

Why is it important to prove f=0 on [a,b] with continuous nonnegative f?

Proving f=0 on [a,b] with continuous nonnegative f is important because it can help in solving differential equations, optimization problems, and other mathematical applications. It also ensures the accuracy and validity of mathematical models and calculations.

What are the steps to prove f=0 on [a,b] with continuous nonnegative f?

The steps to prove f=0 on [a,b] with continuous nonnegative f are:

  1. Show that f(x)=0 for at least one value of x in the interval [a,b].
  2. Prove that f(x) remains nonnegative for all values of x in the interval [a,b].
  3. Show that f(x) is continuous throughout the interval [a,b].

What are some techniques that can be used to prove f=0 on [a,b] with continuous nonnegative f?

Some techniques that can be used to prove f=0 on [a,b] with continuous nonnegative f are:

  • Direct proof: directly showing that f(x)=0 for all x in [a,b] using algebraic manipulations or substitution.
  • Proof by contradiction: assuming that f(x) is not equal to 0 and then showing that this leads to a contradiction.
  • Proof by contrapositive: proving that if f(x) is not equal to 0, then it is not continuous or nonnegative.
  • Proof by induction: using mathematical induction to prove that f(x)=0 for all values of x in [a,b].

Can f=0 on [a,b] be proven with discontinuous or negative f?

No, f=0 on [a,b] can only be proven with continuous and nonnegative f. If f is discontinuous or negative, it cannot be equal to 0 for all values of x in the interval [a,b]. This would violate the definition of continuity and nonnegativity.

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