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Show that $a x^{2}+b x+c$ always has the same sign as the constant term $c$ if $b^{2}-4 a c<0$.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

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Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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I want to show that X squared plus bx plus C always has the same sign as the constant term. See if b squared minus four A. C is less than zero. So what we end up seeing is that if we have a squared X squared plus bx plus C, we know it's going to have the same sign as the constant term. That's because when we have eggs squared plus bx plus C, we end up having. Is that for the quadratic formula we know that X is equal to negative, be closer known as the square root of b squared minus four A. C. All divided by two a. Exactly. Yeah. So we find is that if We squared -4, ac is less than zero, this is negative, so makes this imaginary. And if this is imaginary then we see it's always the same sign as the constant of C. So it's our final answer.

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