Question

If $g(x)=x^{2}-x, x\in\mathbb{R},$ then $g(x)$ is increasing in ________.

• A. $(-\infty,\infty)$
• B. $(-\infty,0)$
• C. $(0,-\infty)$
• D. (-5,5)
• E. $(\frac{1}{2},\infty)$

Hint:

To find increasing intervals, solve $f'(x) > 0$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
A function is said to be increasing on an interval if its derivative is positive on that interval. We need to find the first derivative of the function \(g(x)\), set it to be greater than zero, and solve the resulting inequality for \(x\).
Step 2: Key Formula or Approach
A differentiable function \(g(x)\) is increasing on an interval \(I\) if \(g'(x)>0\) for all \(x \in I\).
1. Find the derivative \(g'(x)\).
2. Solve the inequality \(g'(x)>0\).
Step 3: Detailed Explanation
1. Find the derivative of g(x).
The function is \(g(x) = x^2 - x\).
Using the power rule for differentiation:
\[ g'(x) = \frac{d}{dx}(x^2 - x) = 2x - 1 \] 2. Find the interval where the function is increasing.
We need to find where \(g'(x)>0\).
\[ 2x - 1>0 \] Add 1 to both sides:
\[ 2x>1 \] Divide by 2:
\[ x>\frac{1}{2} \] 3. Express the solution in interval notation.
The function is increasing for all values of \(x\) greater than \(\frac{1}{2}\). In interval notation, this is:
\[ \left(\frac{1}{2}, \infty\right) \] Step 4: Final Answer
The function \(g(x)\) is increasing in the interval \((\frac{1}{2}, \infty)\).