find the stationary points of the following f(x)=x^3-3x^2-24x-7
To find the stationary points of a function, you need to find the values of x where the derivative of the function is equal to zero. The stationary points correspond to the points on the graph where the slope is zero, which can be a maximum, a minimum, or an inflection point.
Let's find the derivative of the given function, f(x) = x^3 - 3x^2 - 24x - 7:
f'(x) = 3x^2 - 6x - 24
To find the stationary points, we need to solve the equation f'(x) = 0:
3x^2 - 6x - 24 = 0
First, we can simplify the equation by dividing all terms by 3:
x^2 - 2x - 8 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. In this case, the equation factors nicely:
(x - 4)(x + 2) = 0
From this equation, we get two possible values for x:
x = 4 and x = -2
So, the stationary points of the function f(x) = x^3 - 3x^2 - 24x - 7 are at x = 4 and x = -2.