How do you find critical points in a function? For example, find the critical points in f(x)=12+2x^2-x^4
please help me somebody
find the derivative and set it equal to zero
To find the critical points of a function, you need to find the values of x that make the derivative of the function equal to zero. In this case, let's find the derivative of the function f(x) = 12 + 2x^2 - x^4.
Step 1: Find the derivative f'(x) of the function f(x).
To find the derivative, you need to apply the power rule and the constant rule. When you take the derivative of a constant, it becomes zero. When you take the derivative of x raised to a power, you bring down the power and multiply it with the coefficient.
f'(x) = 0 + 4x^1 - 4x^3
f'(x) = 4x - 4x^3
Step 2: Set f'(x) equal to zero and solve for x.
To find the critical points, set the derivative f'(x) equal to zero and solve the resulting equation.
4x - 4x^3 = 0
Factoring out 4x from both terms:
4x(1 - x^2) = 0
Setting each factor equal to zero:
4x = 0 ===> x = 0
1 - x^2 = 0
For 1 - x^2 = 0, you can solve it as follows:
Add x^2 to both sides:
1 = x^2
Taking the square root of both sides:
±√1 = x
So, x = ±1
Therefore, the critical points of the function f(x) = 12 + 2x^2 - x^4 are x = -1, 0, and 1.