Question

Find the critical value(s) of the function f(x)=16 x ln x.
x₀=___

Answer 1

The critical value of the function f(x) = 16x ln(x) is found by differentiating, setting the derivative equal to zero, and solving for x, yielding a critical value of approximately 0.368.

Step-by-step explanation:

The critical value(s) of the function f(x) = 16x ln(x) are found by taking the derivative of the function and setting it equal to zero. This is because critical values occur where the derivative is zero or does not exist, corresponding to possible local maxima, minima, or points of inflection on the graph of f(x).

To find the derivative of f(x), we use the product rule:

1. Let u = 16x and v = ln(x). Then f'(x) is u'v + uv'.
2. Derivative of u is 16, and derivative of v is 1/x.
3. So, f'(x) = 16 * ln(x) + 16x * (1/x) = 16 * ln(x) + 16.
4. Setting f'(x) equal to zero gives 16 * ln(x) + 16 = 0. Solving this leads to ln(x) = -1.
5. To solve for x, we take the exponential function of both sides, giving e^(-1) = x.
6. Thus, x is approximately 0.368, which is the critical value for f(x).