How do you find the critical points of #f(x)=(x-10)^2(x+5)#?

Question

Ezekiel Alexander · Accepted Answer

#x = 0, x = 10# are criticial points

As implied by the subtitle of this section, the critical points are those at which the first derivative of an expression are equal to zero. #f(x) = (x−10)^2(x+5)# First, let's expand this: #f(x) = (x^2 −20x +100)(x+5)# #f(x) = x^3 −20x^2 +100x + 5x^2 − 100x +500# #f(x) = x^3 − 15x^2 + 500# NOW we can find the derivative: #f'(x) = 3x^2 - 30x = 0# are the critical points. #x = 0, x = 10# are criticial points

Adam Adkins · Answer

undefined To find the critical points of the function $ f(x) = (x - 10)^2(x + 5) $, follow these steps:

1. Calculate the first derivative of the function $ f'(x) $.
2. Set $ f'(x) = 0 $ and solve for $ x $ to find the potential critical points.
3. Determine whether each potential critical point is a critical point or not by analyzing the behavior of the derivative around these points.

Let's find the critical points:

1. $ f'(x) = 2(x - 10)(x + 5) + (x - 10)^2 $.
2. Set $ f'(x) = 0 $:
   $ 2(x - 10)(x + 5) + (x - 10)^2 = 0 $.
   Solve for $ x $ to find potential critical points.
3. After solving for $ x $, you'll get potential critical points. To determine if they are indeed critical points, you need to analyze the behavior of the derivative around these points. This can involve using the first or second derivative test or looking at the sign changes in the derivative.

Once you've identified the critical points, you can further analyze the function's behavior around these points to understand its local maxima, minima, or inflection points.