Find the Ascending Area of f(x)=-3x²+12: Quadratic Function Analysis

Video Solution

Solution Steps

00:00 Find the domain of increase of the function

00:04 Let's examine the trinomial coefficients

00:08 Let's look at the coefficient of X squared, negative so the function is concave down

00:16 We'll use the formula to find the vertex of the parabola

00:21 We'll substitute appropriate values according to the given data and solve to find the vertex

00:25 This is the X value at the vertex of the parabola

00:28 Let's determine when the parabola is decreasing and when it's increasing based on its type

00:31 Let's draw the X-axis and find the domain of increase

00:38 And this is the solution to the question

Step-by-Step Solution

Let's solve the problem.

Step 1: Calculate the derivative of the function:

The function is $f(x) = -3x^2 + 12$ .

The derivative $f'(x)$ is calculated using the power rule:

$f'(x) = \frac{d}{dx}(-3x^2 + 12) = -6x$ .

Step 2: Find where the derivative is positive:

To find the interval where the function is increasing, solve the inequality $f'(x) > 0$ .

This yields:

$-6x > 0$

Divide both sides by $-6$ (remember to reverse the inequality sign when dividing by a negative):

$x < 0$

Therefore, the function is increasing for $x < 0$ .

Thus, the ascending area of the function is $x < 0$ .