Question

Find all critical points of the function f(x)=x ³ −4x ² −4x−10.f there is more than one critical point, list them in descending order and separate them by commas.

A) x=−2
B) x=−1
C) x=1
D) x=2

Answer 1

To determine the critical points of the function f(x) = x³ - 4x² - 4x - 10, we must find and solve the derivative set to zero. Using the quadratic formula, we can identify the critical points which should then be listed in descending order. The correct answer is A) x=−2

Step-by-step explanation:

Finding Critical Points of a Function

To find the critical points of the function f(x) = x³ - 4x² - 4x - 10, we first need to find its derivative which will give us the rate of change of the function. Critical points occur when this derivative is zero or undefined, indicating a potential maximum, minimum, or inflection point.

The derivative of the given function is f'(x) = 3x² - 8x - 4. Setting the derivative equal to zero gives us a quadratic equation:

3x² - 8x - 4 = 0

To solve this quadratic equation, we can use the quadratic formula, which is given by:

x = (-b ± √(b² - 4ac)) / (2a)

In our case, a = 3, b = -8, and c = -4. Plugging these values into the quadratic formula will give us the critical points of the function.

Once the critical points are found, they should be listed in descending order, separated by commas. The options given in the question are A) x = -2, B) x = -1, C) x = 1, and D) x = 2, which we will need to verify with our calculated critical points.