f(x)=2ln(x^2+3)-x Domain [-3,5]
how do i find the absolute maximum value of this?
To find the absolute maximum value of the function f(x) = 2ln(x^2 + 3) - x within the given domain [-3, 5], you need to follow these steps:
1. Start by finding the critical points of the function within the given domain. Critical points occur when the derivative of the function is either zero or undefined.
2. Calculate the derivative of f(x) using the properties of logarithmic functions. The derivative of ln(u) with respect to x is given by (1/u) * du/dx. Applying this property, the derivative of f(x) becomes:
f'(x) = (2 / (x^2 + 3)) * (2x) - 1
Simplifying, we get:
f'(x) = 4x / (x^2 + 3) - 1
3. Set the derivative equal to zero and solve for x to find the critical points. In this case, solve the equation:
4x / (x^2 + 3) - 1 = 0
This expression simplifies to:
4x = x^2 + 3
Rearranging the terms:
x^2 - 4x + 3 = 0
Factorizing this quadratic equation, we get:
(x - 1)(x - 3) = 0
Thus, the critical points within the given domain are x = 1 and x = 3.
4. Evaluate the function f(x) at the critical points and the endpoints of the given domain to identify the maximum value. We have the following values to consider:
f(-3), f(1), f(3), and f(5)
To find these values, substitute the corresponding x-values into the function f(x):
f(-3) = 2ln((-3)^2 + 3) - (-3)
f(1) = 2ln(1^2 + 3) - 1
f(3) = 2ln(3^2 + 3) - 3
f(5) = 2ln(5^2 + 3) - 5
Evaluate each of the above expressions to calculate the corresponding values.
5. Finally, compare the values obtained in step 4 and determine the maximum value among them. That value will be the absolute maximum value of the function f(x) within the given domain [-3, 5].
Note: Alternatively, you can graph the function on a calculator or computer software to visualize the shape of the curve and identify the absolute maximum on the graph.