4.
f(x)= 7-5x^4
What does x equal? what is the absoloute max?
To find the value of x that makes f(x) equal to a certain value or find the absolute maximum, we'll need to take the derivative of the function and find the critical points.
1. Find the derivative of f(x) using the power rule:
f'(x) = -20x^3
2. Set the derivative equal to zero to find the critical points:
-20x^3 = 0
To solve this equation, we can divide both sides by -20:
x^3 = 0
To solve for x, we can take the cube root of both sides:
x = 0
So, the critical point is x = 0.
3. To determine whether this critical point is a maximum or minimum, we can use the second derivative test.
Find the second derivative f''(x) by taking the derivative of f'(x):
f''(x) = -60x^2
4. Substitute the critical point x = 0 into the second derivative:
f''(0) = -60(0)^2 = 0
Since the second derivative is zero, the second derivative test is inconclusive.
5. To find the absolute maximum (if it exists), we need to evaluate the function at the critical points and the endpoints of the domain. However, since there are no specific bounds or domain mentioned in the question, we'll assume the domain is all real numbers.
Evaluate f(x) at x = 0:
f(0) = 7 - 5(0)^4 = 7
Therefore, the function f(x) has an absolute maximum of 7 at x = 0.