Determine whether the statement is true or false.
d
dx |x2 + x| = |2x + 1|
To determine whether the statement is true or false, we need to solve the equation and check if it holds for all values of x.
Let's solve the equation:
dx |x^2 + x| = |2x + 1|
Since there is a derivative involved, let's assume that d refers to the derivative with respect to x.
For the function |x^2 + x|, we need to consider two cases:
Case 1: (x^2 + x) > 0
In this case, the function |x^2 + x| simplifies to x^2 + x, and the equation becomes:
dx (x^2 + x) = |2x + 1|
Differentiating the function x^2 + x with respect to x, we get:
2x + 1 = |2x + 1|
Since the absolute value of (2x + 1) is equal to (2x + 1), the equation holds in this case.
Case 2: (x^2 + x) < 0
In this case, the function |x^2 + x| simplifies to -(x^2 + x), and the equation becomes:
dx (-(x^2 + x)) = |2x + 1|
Differentiating the function -(x^2 + x) with respect to x, we get:
-2x - 1 = |2x + 1|
For this case, the absolute value of (2x + 1) is equal to -(2x + 1) since (2x + 1) < 0. Therefore, the equation holds in this case as well.
Since the equation holds for both cases, the statement is true.
To determine whether the statement is true or false, we need to simplify the expression on both sides of the equation and see if they are equal.
Let's start by simplifying the left side:
dx |x^2 + x|
The absolute value of any number or expression is always non-negative, so we can rewrite it as follows:
dx (x^2 + x) [Note: We can remove the absolute value here since x^2 + x is always non-negative]
Now, let's simplify the right side:
|2x + 1|
The absolute value of any number or expression is always non-negative, so we can rewrite it as follows:
2x + 1 [Note: We can remove the absolute value here since 2x + 1 is always non-negative]
So now we have the equation as:
dx (x^2 + x) = 2x + 1
To confirm if the statement is true or false, we need to solve this equation. In this case, we have a variable dx multiplying a quadratic equation.
To find the solution, we need more information about the value of dx. If dx is a constant (a fixed number), we can solve the equation by expanding the expression on the left side, simplifying, and finding the values of x that satisfy the equation.
If dx is a variable, we would need additional information or context to determine if the statement is true or false.
Therefore, without more information about the value of dx, we cannot definitively determine whether the statement is true or false.
Determine on what intervals you have that:
|x^2 + x | = x^2 + x
and where do you have that:
|x^2 + x | = -(x^2 + x)
You can then compute the derivative on these intervals and see if what you get is the same as |2x + 1| for all x.