How would you write open intervals for concavity? After you find the second derivative of a fuction; How would you go about finding the concavity
concave up when f"(x) > 0
I understand that. But when writing the concavity intercals do you use the critical numbers or the points choosen around the critical numbers to denote the up and down? For instance This question
Determin the open intervals on which the graph of of 3x^2+7x-3 is concve up or concve down and I have the derivative as 6x-7 and the crtical number as -7/6; where do I go from there?
critical numbers have nothing to do with concavity. They are used for determining min/max, or where f(x) is undefined.
For your f(x) above, f"(x) = 6, which is always positive.
Makes sense, since the curve is a parabola which opens upward, and is thus always concave up.
Don't forget your algebra I when using calculus.
To determine the concavity of a function, after finding the second derivative, you need to analyze the sign of the second derivative over various intervals. Below are the steps to find the concavity of a function:
1. Differentiate the function twice to obtain the second derivative.
2. Simplify the second derivative expression, if needed.
Now, to write open intervals for concavity, follow these steps:
1. Set the second derivative expression equal to zero.
2. Solve the equation to find the critical points or values at which the second derivative changes sign.
3. The values obtained in step 2 divide the domain of the function into intervals.
4. Choose a test point from each interval.
5. Substitute the test point into the second derivative expression and evaluate its sign.
- If the result is positive, the function is concave up in that interval.
- If the result is negative, the function is concave down in that interval.
Repeat steps 4 and 5 for all intervals. The intervals where the sign of the second derivative changes will be points of inflection. Remember that open intervals are expressed using parentheses.
For example, let's say the second derivative is:
f''(x) = 6x - 4
To find the open intervals for concavity, follow these steps:
1. Set 6x - 4 = 0.
6x = 4
x = 4/6 = 2/3
So, x = 2/3 is the critical point.
Now, consider two intervals:
Interval 1: (-∞, 2/3)
Interval 2: (2/3, +∞)
Now, choose a test point from each interval:
- For Interval 1: Pick x = 0
- For Interval 2: Pick x = 1
Substitute these test points into the second derivative expression:
- f''(0) = 6(0) - 4 = -4 (negative)
- f''(1) = 6(1) - 4 = 2 (positive)
Based on the signs of the test points, we can conclude:
- In Interval 1, the function is concave down (negative second derivative).
- In Interval 2, the function is concave up (positive second derivative).
Therefore, the open intervals for concavity are (-∞, 2/3) and (2/3, +∞).