Increasing/decreasing f(x)

  • Thread starter Kamataat
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In summary, the function y=x^2 decreases for values of x less than zero and increases for values of x greater than zero, but the value of x=0 does not fall into either category as the slope is zero at that point. Similarly, for the function y=x^3, it is increasing for values of x less than zero and for values greater than zero, but the definition of "strictly increasing" refers to the entire interval and does not take into account individual points such as x=0 where the slope is zero. This can be confusing, but it ultimately depends on the specific definition and context being used.
  • #1
Kamataat
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Let's take [itex]y=x^2[/itex] as an example. This function decreases if [itex]-\infty < x < 0[/itex] and increases if [itex]0 > x > \infty[/itex]. But what about [itex]x=0[/itex]? Shouldn't it be included in one of the two ranges of [itex]x[/itex]?

- Kamataat
 
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  • #2
Why do you think it should ? This boils down to whether you want zero to be "both positive and negative" or "neither positive nor negative". The latter is more useful.
 
  • #3
So the function [itex]y=x^3[/itex] is not increasing if [itex]-\infty < x < \infty[/itex] (1), but instead is increasing if [itex]-\infty < x < 0[/itex] and if [itex]0 < x < \infty[/itex], because [itex]y'(0)=0[/itex] cuts the range (1) in two pieces?

- Kamataat
 
  • #4
Kamataat said:
But what about [itex]x=0[/itex]? Shouldn't it be included in one of the two ranges of [itex]x[/itex]?
Why? At x = 0 the slope is zero: y is neither increasing nor decreasing.
 
  • #5
Yes, I know that, but my confusion arises from my textbook saying that [itex]y=x^3[/itex] is a strictly increasing function for all [itex]x \in X[/itex]. How can that be right, if at one x (namely x=0), y'=0 and the function is thus constant?

- Kamataat
 
  • #6
strictly increasing

I think it hinges on the definition of "strictly increasing", which is based on an interval: f(x) is strictly increasing if a < b implies f(a) < f(b).

See: http://planetmath.org/encyclopedia/IncreasingdecreasingmonotoneFunction.html [Broken] & http://www.mathreference.com/ca,inc.html
 
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  • #7
yup, that's how it's defined in the book, but it still confuses me. according to the definition of "strictly increasing", i'd say that x^3 is strictly increasing for all x, but then we have the definition that a function is neither decreasing nor incresing if y'=0 (which is true for x^3). so the second def says that x^3 is not increasing for ALL x.

- Kamataat
 
  • #8
nevermind, the 2nd link seems to explain it. thank you!

- Kamataat
 

1. How can I increase the value of f(x)?

To increase the value of f(x), you can either add a positive constant to the function or multiply it by a number greater than 1. This will shift the graph of the function upwards.

2. What effect will decreasing f(x) have on its graph?

Decreasing f(x) will shift the graph downwards. This can be achieved by subtracting a positive constant from the function or multiplying it by a number between 0 and 1.

3. Can I increase f(x) without changing the shape of its graph?

Yes, you can increase f(x) without changing the shape of its graph by adding a constant to the function. This will shift the graph upwards but maintain its original shape.

4. How does increasing/decreasing the coefficient of x affect f(x)?

The coefficient of x affects the slope of the graph of f(x). Increasing the coefficient will make the graph steeper, while decreasing it will make it flatter. This will also affect the rate at which the function increases or decreases.

5. Can I use calculus to increase/decrease f(x)?

Yes, calculus can be used to increase or decrease f(x) by finding the derivative of the function and setting it equal to 0. This will give the critical points of the function where it either reaches a maximum or minimum value. By adjusting the function at these points, you can increase or decrease its value.

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