First, second and third derivatives of a polynomial

In summary, the first derivative of a polynomial is the slope of the tangent line, the second derivative is the concavity, and the third derivative is the rate of change of the concavity. To find these derivatives, you can use various rules such as the power rule, product rule, and quotient rule. These derivatives provide important information about the behavior of a polynomial function, such as slope, concavity, and rate of change, which are useful in real-world applications in mathematics and science.
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Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for every real $x$, then $p(x)\ge 0$ for every real $x$.
 
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If $f(x)$ is a polynomial with real coefficients then $f(x)\to +\infty$ as $x\to\infty$ if the coefficient of its leading term is positive, and $f(x)\to -\infty$ as $x\to\infty$ if the coefficient of its leading term is negative. The degree of the derivative $f'(x)$ is lower than the degree of $f(x)$, so the polynomials $f(x)$ and $f(x) - f'(x)$ have the same leading term. Therefore either they both go to $+\infty$ or they both go to $-\infty$, as $x\to\infty$.

Now suppose that $f(x) - f'(x) \geqslant0$ for every real $x$. Then $f(x) - f'(x)\to +\infty$ as $x\to\infty$ and therefore $f(x)\to +\infty$ as $x\to\infty$. By the same argument, $f(x)\to +\infty$ as $x\to-\infty$. It follows that $f(x)$ has a finite minimum value, which it attains at some point $x_0$. Then $f'(x_0) = 0$. But $f(x_0) - f'(x_0) \geqslant0$. So $f(x_0)\geqslant0$. But if the minimum value of $f(x)$ is nonnegative then the function must be nonnegative everywhere. That proves
Fact 1: If $f(x)$ is a polynomial with real coefficients, and $f(x) - f'(x) \geqslant0$ for every real $x$, then $f(x)\geqslant0$ for every real $x$.
It follows that
Fact 2: If $f(x)$ is a polynomial with real coefficients, and $f(x) +f'(x) \geqslant0$ for every real $x$, then $f(x)\geqslant0$ for every real $x$.
To prove Fact 2, let $g(x) = f(-x)$. Then $g'(x) = -f'(-x)$ and so $g(x) - g'(x) = f(-x) + f'(-x) \geqslant0$ for every real $x$. It follows from Fact 1 that $g(x)\geqslant0$ for every real $x$. Therefore $f(x)\geqslant0$ for every real $x$.

Write $D= \frac d{dx}$ for the operator of differentiation, and $I$ for the identity operator. Then those two Facts can be written as
Fact 1: If $f(x)$ is a polynomial with real coefficients, and $(I-D)f(x) \geqslant0$ for every real $x$, then $f(x)\geqslant0$ for every real $x$;
Fact 2: If $f(x)$ is a polynomial with real coefficients, and $(I+D)f(x) \geqslant0$ for every real $x$, then $f(x)\geqslant0$ for every real $x$.

Now let $p(x)$ be a polynomial with real coefficients such that $p(x) - p'(x) - p''(x) + p'''(x) \geqslant0$ for every real $x$. That condition says that $(I-D - D^2 + D^3)p(x)\geqslant0$ for every real $x$. But $I-D-D^2+D^3 = (I-D)(I-D^2)$. Therefore $(I-D - D^2 + D^3)p(x) = (I-D)\bigl((I-D^2)p(x)\bigr) \geqslant0$ for every real $x$. It follows from Fact 1 that $(I-D^2)p(x) \geqslant0$ for every real $x$. But $I-D^2 = (I-D)(I+D)$. Therefore $(I-D)\bigl((I+D)p(x)\bigr) \geqslant0$ for every real $x$, and from Fact 1 again it follows that $(I+D)p(x) \geqslant0$ for every real $x$. Finally, by Fact 2 it then follows that $p(x) \geqslant0$ for every real $x$.
 

FAQ: First, second and third derivatives of a polynomial

1. What is a polynomial?

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. It can have one or more terms, and the exponents must be whole numbers.

2. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is calculated by finding the slope of the tangent line to the function at that point.

3. What is the first derivative of a polynomial?

The first derivative of a polynomial is the derivative of the polynomial function. It represents the slope of the tangent line to the polynomial at any given point.

4. What is the second derivative of a polynomial?

The second derivative of a polynomial is the derivative of the first derivative. It represents the rate of change of the slope of the polynomial at any given point.

5. What is the third derivative of a polynomial?

The third derivative of a polynomial is the derivative of the second derivative. It represents the rate of change of the rate of change of the slope of the polynomial at any given point.

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