- #1
dirk_mec1
- 761
- 13
Homework Statement
http://img143.imageshack.us/img143/1861/14726023et5.png
Can I take c such that
[tex]
c = \frac{2 ||f ||_{\infty} }{ ||f||_E}
[/tex]
and in case f= 0 everywhere c=1?
Last edited by a moderator:
Yes, you're right!Dick said:The point to c being a constant is that it's independent of f, right?
Try and read the question eliminating the 'norm' words. You have a function f(x) that has a bounded derivative on [0,1] and f(0)=0. Can you get a bound for f(x) in terms of the bound on the derivative?
Dick said:If you are studying Banach spaces, you must know more about derivatives than just the definition. If you know f'(x) how can you find f(x)?
Take the absolute value of both sides of this, and then work from there.dirk_mec1 said:[tex]f(x) = \int_0^x f'(t)\ \mbox{d}t [/tex]
dirk_mec1 said:Do you mean like this?
[tex]|f(x)| = |\int_0^x f'(t)\ \mbox{d}t| \leq \int_0^x |f'(t)|\ \mbox{d}t \leq \int_0^x |f'(t)|_{\infty} \ \mbox{d}t \leq \int_0^1 M \mbox{d}t = M
[/tex]
Is this correct?
Solving for c refers to finding the value of the variable "c" in an equation or mathematical problem. It is a common practice in mathematics and science to solve for a specific variable in order to find a solution or understand the relationship between different variables.
No, it is not always possible to solve for c. In some cases, the equation or problem may have multiple solutions or no solution at all. It depends on the complexity of the problem and the given information. However, in many cases, it is possible to solve for c by using different mathematical techniques or by rearranging the equation.
Some common methods used to solve for c include substitution, elimination, graphing, and using algebraic properties such as the distributive property or combining like terms. The method used will depend on the type of equation or problem and the given information.
Sure, let's say we have the equation 3c + 9 = 24. To solve for c, we can use the algebraic property of subtraction to isolate the variable c on one side of the equation. First, we subtract 9 from both sides of the equation, which gives us 3c = 15. Then, we divide both sides by 3 to get the value of c, which is 5. So, in this case, we have successfully solved for c.
Solving for c is a fundamental skill in mathematics and science that can be applied to various real-life situations. For example, engineers use it to design structures and machines, economists use it to analyze market trends, and chemists use it to calculate chemical reactions. In everyday life, solving for c can also help with budgeting, planning, and problem-solving.