f(x) = 9 sqrt(x)e*(-x)

I will assume you meant

y = 9x^(1/2) (e^-x)
dy/dx = 9x^(1/2) (-e^-x) + (e^-x)(9/2)x^(-1/2)

= -(9/2)x^(-1/2) (e^-x)[2x - 1]
= 0 for a max/min of y

if -(9/2)x^(-1/2)=0 we have no solution
if e^-x = 0 we have no solution
if 2x-1 = 0 we get x = 1/2

f(1/2) = 9√(1/2) e^(-1/2) = appr. 3.86

To find the derivative of the function f(x) = 9 sqrt(x)e^(-x), we can use the product rule and the chain rule.

Step 1: Separate the function into two parts: g(x) = 9 sqrt(x) and h(x) = e^(-x).

Step 2: Find the derivative of g(x) using the power rule and the chain rule. The power rule states that the derivative of x^n is n*x^(n-1).

g'(x) = 9 * (1/2) * x^(-1/2)
= 9/2 * x^(-1/2)

Step 3: Find the derivative of h(x) using the chain rule. The chain rule states that the derivative of e^u, where u is a function of x, is e^u times the derivative of u with respect to x.

h'(x) = e^(-x) * (-1)
= -e^(-x)

Step 4: Apply the product rule to find the derivative of f(x). The product rule states that the derivative of fg is f'g + fg'.

f'(x) = g'(x) * h(x) + g(x) * h'(x)
= (9/2 * x^(-1/2)) * e^(-x) + (9 sqrt(x)) * (-e^(-x))

Simplifying the expression, we can rewrite it as:

f'(x) = (9/2 * x^(-1/2) - 9 sqrt(x)) * e^(-x)

Therefore, the derivative of f(x) = 9 sqrt(x)e^(-x) is f'(x) = (9/2 * x^(-1/2) - 9 sqrt(x)) * e^(-x).

The given function is f(x) = 9sqrt(x)e^(-x).

To understand and analyze this function, let's break it down:

1. The term 'sqrt(x)' represents the square root of x. For example, sqrt(4) = 2, sqrt(9) = 3, etc.
2. The letter 'e' represents Euler's number, which is approximately equal to 2.71828. It is often used in mathematical calculations and is the base of natural logarithms.
3. The exponent '-x' indicates that the function is being raised to the power of negative x. In other words, it is multiplied by 1/e^x.

Now, let's discuss how to evaluate this function for a specific value of x.

To find the value of f(x) for a given x, follow these steps:

1. Replace the variable x in the function f(x) with the specific value you want to evaluate. For example, if you want to find f(2), substitute x = 2 into the function:
f(2) = 9sqrt(2)e^(-2).

2. Calculate the square root of 2:
sqrt(2) ≈ 1.41421356.

3. Evaluate the exponent:
e^(-2) ≈ 0.135335.

4. Multiply the square root, exponent, and 9 together:
f(2) ≈ 9 * 1.41421356 * 0.135335 ≈ 1.63632.

Therefore, f(2) is approximately 1.63632.

Similarly, you can evaluate f(x) for any other value of x by following these steps.