Find the value of the derivative at each indicated extremum.
what is the derivative of -3x((x+1)^1/2)?
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To find the value of the derivative at each indicated extremum, we first need to find the extremum points of the function.
The extremum points are the points on the graph where the derivative changes signs or equals zero.
To find the extremum points, we can follow these steps:
1. Calculate the derivative of the function.
2. Set the derivative equal to zero and solve for x.
3. Plug the x-values obtained in Step 2 into the derivative to find the corresponding y-values.
Let's go through an example to illustrate this process:
Example:
Suppose we have the function f(x) = x^3 - 3x^2 + 2x. We want to find the value of the derivative at each indicated extremum.
Step 1: Calculating the derivative
To find the derivative of f(x), we can use the power rule of differentiation:
f'(x) = 3x^2 - 6x + 2
Step 2: Setting the derivative equal to zero
Setting f'(x) = 0, we get:
3x^2 - 6x + 2 = 0
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
Step 3: Finding the extremum points
Let's suppose we solve the equation and obtain two x-values: x1 and x2.
Next, we can plug these x-values back into the derivative f'(x) to find the corresponding y-values:
f'(x1) = 3(x1)^2 - 6(x1) + 2
f'(x2) = 3(x2)^2 - 6(x2) + 2
These resulting values will give us the value of the derivative at each indicated extremum.
Keep in mind that if the extremum points are maximum or minimum points, the value of the derivative at those points will be zero. However, if the extremum point is an inflection point, the value of the derivative will not be zero.