Calculate the Descending Area of f(x) = (1/2)x²: Quadratic Function Analysis

Video Solution

Solution Steps

00:00 Find the domain of decrease of the function

00:03 Notice the coefficient of X squared is positive, so the function is concave up

00:07 Let's look at the trinomial coefficients

00:15 Use the formula to find the vertex of the parabola

00:21 Substitute appropriate values according to the given data and solve to find the vertex

00:24 This is the X value at the vertex of the parabola

00:29 Determine when the parabola decreases and increases based on its type

00:33 Draw the X-axis and find the domain of decrease

00:42 And this is the solution to the question

Step-by-Step Solution

To solve the problem of finding the descending area of the function $f(x) = \frac{1}{2}x^2$ , we follow these steps:

Step 1: Calculate the derivative of the given function. The function is $f(x) = \frac{1}{2}x^2$ . Differentiating this, we get $f'(x) = \frac{d}{dx}(\frac{1}{2}x^2) = x$ .
Step 2: Determine where the derivative is negative. Since $f'(x) = x$ , the derivative is negative when $x < 0$ .
Step 3: Conclude the solution. We find that the function $f(x)$ is decreasing for $x < 0$ .

Thus, the descending area (domain where the function is decreasing) for the function $f(x) = \frac{1}{2}x^2$ is $x < 0$ .

The correct choice that matches this solution is: $x < 0$ .