Find values for a and b so the function f(x)= ax*e^bx has the max value f(1)=3. Give exact values for a and b, not decimals)
To find the values for a and b that maximize the function f(x) = ax * e^(bx) given that f(1) = 3, we need to follow these steps:
Step 1: Calculate the derivative of the function f(x) with respect to x, denoted as f'(x). This derivative will help us find the critical points of the function.
Step 2: Set f'(x) = 0 and solve for x to find the critical points.
Step 3: Plug the critical point(s) into the original function f(x) to determine the corresponding y-values.
Let's begin step by step:
Step 1: Calculate the derivative of f(x) = ax * e^(bx) with respect to x, denoted as f'(x).
Using the product rule and chain rule of differentiation, we obtain:
f'(x) = a * (e^(bx)) * x * b + ax * (e^(bx)) * b
Simplifying this expression, we get:
f'(x) = abx * e^(bx) + abx^2 * e^(bx)
Step 2: Set f'(x) = 0 and solve for x to find the critical points.
Setting f'(x) = 0:
abx * e^(bx) + abx^2 * e^(bx) = 0
Since e^(bx) is always positive, we can divide the equation by abx * e^(bx) to simplify it:
x + bx = 0
Factoring out x, we get:
x(1 + b) = 0
Therefore, we have two critical points:
1) x = 0
2) 1 + b = 0, which implies b = -1
Step 3: Plug the critical points into the original function f(x) to determine the corresponding y-values.
For x = 0:
f(0) = a(0) * e^(b(0)) = 0 * e^0 = 0
For b = -1:
f(1) = a(1) * e^(-1 * 1) = a * e^(-1)
Given that f(1) = 3, we can write:
a * e^(-1) = 3
To find the exact values for a and b, we need one more equation related to a. Unfortunately, the given information is not sufficient to determine the exact values of a and b. We have an equation relating a and e^(-1), but we cannot determine their exact values without additional information or an additional equation.
Hence, we cannot provide the exact values for a and b without more information.