Calculate Area Under (x+5)²: Finding the Descending Region

Question

Find the descending area of the function

$y=(x+5)^2$

00:00 Find the domain of decrease of the function

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

00:07 Note the coefficient of X squared, positive - smiling function

00:14 Examine the function's coefficients

00:20 Use the formula to find the vertex point

00:31 Substitute appropriate values and solve to find the vertex point

00:40 This is the X value at the vertex point

00:44 In a minimum parabola, the domain of decrease is before the vertex point

00:48 And this is the solution to the question

To solve this problem, we will determine when the function $y = (x+5)^2$ is decreasing by using its derivative:

Step 1: Differentiate $y = (x+5)^2$ with respect to $x$ , yielding $\frac{dy}{dx} = 2(x+5)$ .
Step 2: Set the derivative less than zero: $2(x+5) < 0$ .
Step 3: Simplify the inequality to solve for $x$ :
$x + 5 < 0$ implies $x < -5$ .

The function is decreasing for values of $x$ that satisfy $x < -5$ .

Therefore, the solution is $x < -5$ .

$x < -5$