The critical number of the function

f(x)=6xlnx is x=

f' = 6lnx + 6

f'=0 at x = 1/e

To find the critical number of a function, we need to find the points where the derivative of the function is either zero or undefined.

Let's start by finding the derivative of the function f(x) = 6xln(x). We will use the product rule here.

f'(x) = 6(ln(x) + x * (1/x))
= 6ln(x) + 6

Next, we need to find the values of x where the derivative is either zero or undefined. Setting the derivative equal to zero and solving for x, we get:

6ln(x) + 6 = 0
6ln(x) = -6
ln(x) = -1
x = e^(-1)

This gives us one critical number x = e^(-1).

Now, to check for any points where the derivative is undefined, we need to look for values where the natural logarithm function is undefined. The natural logarithm function, ln(x), is undefined for x ≤ 0.

Since x = e^(-1) is a positive value, we can disregard the possibility of the derivative being undefined.

Therefore, the critical number of the function f(x) = 6xln(x) is x = e^(-1), which is approximately 0.368.

To find the critical number of a function, we need to first calculate its derivative and solve for the values of x where the derivative is either zero or undefined.

Let's start by finding the derivative of the given function f(x) = 6xln(x). We can use the product rule to differentiate this function. The product rule states that if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).

In this case, g(x) = 6x and h(x) = ln(x). Differentiating g(x) with respect to x, we get g'(x) = 6. Differentiating h(x) with respect to x requires the use of the chain rule since we have a composite function.

The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x). In this case, f(u) = ln(u) and u = x. Therefore, h'(x) = (1/u) * g'(x) = (1/x) * 6 = 6/x.

Now, applying the product rule, we can find the derivative of f(x):
f'(x) = g'(x) * h(x) + g(x) * h'(x) = 6x * ln(x) + 6 * (6/x) = 6x ln(x) + 36/x.

To find the critical numbers, we set the derivative equal to zero:
6x ln(x) + 36/x = 0.

To simplify this equation, we can divide through by 6:
x ln(x) + 6/x = 0.

To solve this equation, it can't be solved algebraically, so we need to use numerical methods or a graphing calculator. One possible method is to graph the function f(x) = x ln(x) + 6/x and look for the values of x where the graph intersects the x-axis (where the function equals zero), indicating critical numbers.

By observing the graph or using a graphing calculator, we can see that the function intersects the x-axis at x ≈ 0.645 and x ≈ 7.354. These are the critical numbers of the function f(x) = 6x ln(x).