Prove that this function is nonnegative

Thread starter sergey_le
Start date Jan 19, 2020
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Function

In summary, the conversation discusses how to prove that the function f(x)=##x^2##/2 - xcosx+sinx is greater than 0 for all values of x except 0. The participants suggest using the Mean Value Theorem and a proof by contradiction to show that f(x) is always positive.

Jan 30, 2020

PeroK

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I thought it might be worth adding an outline solution to this.

We need to show that ##f(x) > 0 \ ## when ##x \ne 0##, where ##f(x) = \frac 1 2 x^2 - x\cos x + \sin x##.

Note that ##f## is symmetric about the y-axis (##f(x) = f(-x)##), so it is enough to show that ##f(x) > 0## for ##x > 0##. Note also that ##f(0) = 0##.

##f'(x) = x + x\sin x = x(1 + \sin x)##

Therefore ##f'(x) \ge 0## for ##x > 0##, hence (by the mean value theorem) ##f(x) \ge 0## for ##x > 0##.

Note that ##f'(x) = 0## when ##x = \frac 3 2 \pi + 2n\pi##. In particular, ##f'(x) > 0## for ##x \in (0, \pi]##. Therefore, by the mean value theorem, ##f(x) > 0## for ##x \in (0, \pi]##. And, of course, ##f(\pi) > 0##.

Finally, for ##x > \pi## we have ##f'(x) \ge 0##, hence ##f(x) \ge f(\pi) > 0##; and the result follows.

<h2>1. How do you prove that a function is nonnegative?</h2><p>To prove that a function is nonnegative, you need to show that the output of the function is always equal to or greater than zero for all possible inputs. This can be done by using mathematical techniques such as induction, contradiction, or direct proof.</p><h2>2. Can a function be both nonnegative and negative?</h2><p>No, a function cannot be both nonnegative and negative. A function is considered nonnegative if its output is always equal to or greater than zero. If the output is ever less than zero, the function is considered negative.</p><h2>3. What are some common examples of nonnegative functions?</h2><p>Some common examples of nonnegative functions include polynomials, exponential functions, and trigonometric functions. These functions always have a positive or zero output for all possible inputs.</p><h2>4. How does proving a function is nonnegative relate to real-world applications?</h2><p>Proving that a function is nonnegative is important in various real-world applications, such as in economics, physics, and engineering. In economics, nonnegative functions are used to model revenue, profit, and demand. In physics, nonnegative functions are used to represent energy, distance, and time. In engineering, nonnegative functions are used to model signal strength, power, and voltage.</p><h2>5. Can a function be nonnegative on one interval and negative on another?</h2><p>Yes, a function can be nonnegative on one interval and negative on another. This means that the function's output is equal to or greater than zero for some values of the input, but less than zero for other values of the input. In this case, the function is considered nonnegative only on the interval where its output is always equal to or greater than zero.</p>

1. How do you prove that a function is nonnegative?

To prove that a function is nonnegative, you need to show that the output of the function is always equal to or greater than zero for all possible inputs. This can be done by using mathematical techniques such as induction, contradiction, or direct proof.

2. Can a function be both nonnegative and negative?

No, a function cannot be both nonnegative and negative. A function is considered nonnegative if its output is always equal to or greater than zero. If the output is ever less than zero, the function is considered negative.

3. What are some common examples of nonnegative functions?

Some common examples of nonnegative functions include polynomials, exponential functions, and trigonometric functions. These functions always have a positive or zero output for all possible inputs.

4. How does proving a function is nonnegative relate to real-world applications?

Proving that a function is nonnegative is important in various real-world applications, such as in economics, physics, and engineering. In economics, nonnegative functions are used to model revenue, profit, and demand. In physics, nonnegative functions are used to represent energy, distance, and time. In engineering, nonnegative functions are used to model signal strength, power, and voltage.

5. Can a function be nonnegative on one interval and negative on another?

Yes, a function can be nonnegative on one interval and negative on another. This means that the function's output is equal to or greater than zero for some values of the input, but less than zero for other values of the input. In this case, the function is considered nonnegative only on the interval where its output is always equal to or greater than zero.