Find the ascending area of the function
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Find the ascending area of the function
Let's solve the problem.
Step 1: Calculate the derivative of the function:
The function is .
The derivative is calculated using the power rule:
.
Step 2: Find where the derivative is positive:
To find the interval where the function is increasing, solve the inequality .
This yields:
Divide both sides by (remember to reverse the inequality sign when dividing by a negative):
Therefore, the function is increasing for .
Thus, the ascending area of the function is .
Find the ascending area of the function
\( f(x)=2x^2 \)
The ascending area (or increasing interval) is where the function's values get larger as x increases. It's like going uphill on the graph!
The derivative tells you the slope at any point. When , the slope is positive, meaning the function is going upward.
Memory trick: When you divide or multiply both sides by a negative number, always flip the inequality! Think of it as the negative 'flipping' everything around.
Yes! For this parabola opening downward, you can see it increases on the left side of the vertex. But using derivatives gives you the exact mathematical proof.
When , the function has a critical point - it's neither increasing nor decreasing at that instant. Here, that happens at x = 0.
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