Show that f(x)=x^2+b x+c is decreasing on (-∞,-(b)/(2)) and increasing on (-(b)/(2), ∞)

Name: Problem 54, Chapter 4: Calculus for AP
Uploaded: 2019-06-26T17:23:45-08:00
Duration: 2 min 2 s
Channel: Michelle Ling
Description: Show that f(x)=x^2+b x+c is decreasing on (-∞,-(b)/(2)) and increasing on (-(b)/(2), ∞)

step 1 f of x is equal to x squared plus b x plus c is decreasing when x is less than negative b over 2

Show that f(x)=x^2+b x+c is decreasing on (-∞,-(b)/(2)) and...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial typically expressed in the form ax² + bx + c. These functions have parabolic graphs and exhibit properties such as a vertex, axis of symmetry, and a constant concavity determined by the coefficient a. Understanding quadratic functions is fundamental in both algebra and calculus for studying curves and optimization.

Derivative

The derivative is a tool in calculus that measures the rate at which a function changes. It provides valuable information about the behavior of functions, particularly regarding slopes of tangent lines. For polynomial functions, calculating the derivative helps identify where the function is increasing or decreasing.

Critical Point

A critical point is where the derivative of a function is zero or undefined, indicating potential local maximums, minimums, or inflection points. Identifying these points is crucial for understanding the overall behavior of a function and for segmenting intervals of increasing or decreasing trends.

Monotonicity

Monotonicity describes the behavior of a function in terms of being consistently increasing or decreasing over an interval. By analyzing the sign of the derivative, one can determine the intervals on which the function rises or falls. This concept is essential in optimization and in understanding the structure and shape of a function's graph.

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