Calculate Ascending Area: Finding Area of y=(x+3)²+2x²

Video Solution

Solution Steps

00:00 Find the domain of increase of the function

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

00:11 Collect terms

00:21 Notice the coefficient of X squared, positive - smiling function

00:28 Observe the parabola coefficients

00:41 Use the formula to find the vertex point

00:47 Substitute appropriate values and solve to find the vertex point

00:56 This is the X value at the vertex point

01:00 In a minimum parabola, the domain of increase is after the vertex point

01:03 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we will use the following steps:

Let's proceed with the solution:

Step 1: Differentiate the given function with respect to $x$ :

$y = (x+3)^2 + 2x^2$

Differentiate each term:

$\frac{d}{dx}((x+3)^2) = 2(x+3) \cdot 1 = 2(x+3)$

$\frac{d}{dx}(2x^2) = 4x$

Combine the derivatives:

$y' = 2(x+3) + 4x = 2x + 6 + 4x$

$y' = 6x + 6$

Step 2: Find the values of $x$ where the derivative is greater than zero:

$6x + 6 > 0$

Simplify the inequality:

$6x > -6$

$x > -1$

Step 3: The solution to the inequality tells us the interval where the function is increasing:

The function is increasing wherever $x > -1$ .

Therefore, the correct answer is $-1 < x$ .