Where is this function larger then zero?

In summary: So, we just have to worry about the sign of ##-a*A/(b/c + 1)##. Since ##0 < a, 0 < A, 0 < b, 0 < c##, it follows that ##0 < b/c + 1##, since all terms are positive. Then, multiplying by ##-a*A## reverses the sign, so that ##-a*A/(b/c + 1) < 0##, and thus, ##e^{-a*A/(b/c + 1)} < 1##, and therefore, ##1 - e^{-a*A/(b/c + 1)} > 0##.In summary, the expression 1-exp
  • #1
Ad VanderVen
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Poster has been reminded to show their work on schoolwork problems
Homework Statement
It is given that 0 <A, 0 <a, 0 <b, 0 <c. Under what conditions is then
1-exp (-a * A / (b / c + 1)) greater than zero?
Relevant Equations
1-exp (-a * A / (b / c + 1))
I do not know how to proceed.
 
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  • #2
Ad VanderVen said:
Homework Statement:: It is given that 0 <A, 0 <a, 0 <b, 0 <c. Under what conditions is then
1-exp (-a * A / (b / c + 1)) greater than zero?
Relevant Equations:: 1-exp (-a * A / (b / c + 1))

I do not know how to proceed.
You have to make an effort.
 
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  • #3
$$1-exp(-a*A/(b/c+1))>0$$
$$exp(-a*A/(b/c+1))<1$$
$$ln(-a*A/(b/c+1))<0$$
$$0<-a*A/(b/c+1) $$ and $$-a*A/(b/c+1)<1$$
 
  • #4
Ad VanderVen said:
$$exp(-a*A/(b/c+1))<1$$
$$ln(-a*A/(b/c+1))<0$$

This doesn't look right.
 
  • #5
Sorry, You are right. It should be
$$-a*A/(b/c+1)<0$$
and therefore
$$a*A/(b/c+1)>0$$
and because 0 < a, 0 < b, 0 < c, 0 < A it follows that
$$a*A/(b/c+1)>0$$
is always true.

Am I correct?
 
  • #6
Ad VanderVen said:
Sorry, You are right. It should be
$$-a*A/(b/c+1)<0$$
and therefore
$$a*A/(b/c+1)>0$$
and because 0<a, 0<b, 0<c, )< A it follows that
$$a*A/(b/c+1)>0$$
is always true.

Am I correct?
Yes.
 
  • #7
@PeroK As usual, thanks a lot.
 
  • #8
Ad VanderVen said:
@PeroK As usual, thanks a lot.
There as a quick way. First, the exponential is of the form ##e^{-k}##, where ##k > 0##. This is always less than ##1##, because ##e^k > 1## for ##k > 0##, and ##e^{-k} = 1/e^{k}##.
 

Related to Where is this function larger then zero?

1. What is the definition of a function being larger than zero?

A function is considered to be larger than zero when its output or y-value is greater than zero for all possible inputs or x-values within its domain.

2. How can I determine where a function is larger than zero?

To determine where a function is larger than zero, you can graph the function and look for regions where the graph is above the x-axis. Alternatively, you can set the function equal to zero and solve for the x-values that make the function equal to or greater than zero.

3. Can a function be larger than zero at multiple points?

Yes, a function can be larger than zero at multiple points. This means that there are multiple x-values for which the function's output is greater than zero.

4. What does it mean if a function is never larger than zero?

If a function is never larger than zero, it means that its output or y-values are always equal to or less than zero for all possible inputs or x-values within its domain. In other words, the function never crosses or touches the x-axis.

5. How does the concept of a function being larger than zero relate to real-world applications?

In real-world applications, the concept of a function being larger than zero can be used to find the minimum value of a quantity or to determine when a certain condition is met. For example, in economics, a business may use this concept to determine the minimum number of products they need to sell in order to make a profit.

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