Find the Descending Area of y=(x+5)²+2x: Quadratic Function Analysis

Video Solution

Solution Steps

00:00 Find the domain of decrease of the function

00:03 Use the shortened multiplication formulas to expand the parentheses

00:11 Collect like terms

00:24 Note the coefficient of X squared, positive - smiling function

00:29 Observe the function's coefficients

00:34 Use the formula to find the vertex point

00:44 Substitute appropriate values and solve to find the vertex point

00:51 This is the X value at the vertex point

00:56 In a minimum parabola, the domain of decrease is before the vertex point

01:01 And this is the solution to the question

Step-by-Step Solution

To determine where the function $y = (x+5)^2 + 2x$ is decreasing, we need to follow these steps:

Step 1: Differentiate the Function
The function is $y = (x+5)^2 + 2x$ . First, we expand and simplify it:
$y = (x+5)^2 + 2x = x^2 + 10x + 25 + 2x = x^2 + 12x + 25$ .
Now, compute the derivative:
$y' = \frac{d}{dx} (x^2 + 12x + 25) = 2x + 12$ .
Step 2: Identify Intervals of Decrease
Set the derivative less than zero to find where the function is decreasing:
$2x + 12 < 0$ .
Solve for $x$ :
$2x < -12$
$x < -6$ .

Therefore, the function $y = (x+5)^2 + 2x$ is decreasing for $x < -6$ .

The correct answer is $x < -6$ .