Find the formula for a function of the form y=bxe^(-ax) with a local maximum at (4,12)...
Why did the function go see a therapist? Because it couldn't find its x-pression!
Now, let's find the formula for the function you're looking for. We have a function of the form y = bxe^(-ax), and we know it has a local maximum at (4,12).
First, let's find the derivative of the function. Taking the derivative of y with respect to x, we get:
dy/dx = b(e^(-ax) - axe^(-ax))
To find the local maximum, we need to set the derivative equal to zero:
b(e^(-ax) - axe^(-ax)) = 0
From here, we can solve for a. However, the value of b is still unknown. Without the value of b, we cannot determine the exact formula for the function.
To find the formula for a function of the form y = bxe^(-ax) with a local maximum at (4,12), we need to determine the values of b and a.
A local maximum occurs when the derivative of the function is equal to zero. So let's take the derivative of the function:
y = bxe^(-ax)
To find dy/dx, we'll use the product rule and the chain rule:
dy/dx = b * e^(-ax) + (-abx) * e^(-ax)
Now, we can set the derivative equal to zero and solve for x to find the x-coordinate of the local maximum:
0 = b * e^(-ax) + (-abx) * e^(-ax)
Dividing through by e^(-ax), we get:
0 = b - abx
Simplifying further:
b = abx
Next, we can substitute the coordinates of the local maximum (4,12) into the equation and solve it simultaneously with our previous equation:
12 = ab * 4
From this equation, we can deduce that b/a = 3.
Using this information, we can rearrange the equation b = abx to solve for x:
x = 1/a
Now, we know that x = 4 when (4,12) is a local maximum. Substituting this into the equation, we get:
4 = 1/a
Solving for a, we find that a = 1/4.
Finally, substituting the values of a and b into the original equation (y = bxe^(-ax)), we can determine the formula for the function:
y = 3x * e^(-x/4)
Therefore, the formula for the function with a local maximum at (4,12) is y = 3x * e^(-x/4).
To find the formula for a function of the form y = bxe^(-ax) with a local maximum at (4,12), we can start by setting up the problem.
The local maximum occurs at the point (4, 12), which means that the derivative of the function equals zero at x = 4. So, we need to find the derivative of the function first.
The derivative of y with respect to x can be found using the product rule and the chain rule. Let's calculate it step by step:
1. Find the derivative of bxe^(-ax) with respect to x using the product rule:
dy/dx = b * d/dx(x) * e^(-ax) + x * d/dx(e^(-ax))
= b * (1) * e^(-ax) + x * (-a) * e^(-ax)
= b * e^(-ax) - a * x * e^(-ax)
= e^(-ax)(b - a * x)
Now, we have found the derivative of y with respect to x. Next, we need to find the value of b and a by plugging in the point (4, 12) into the expression we obtained.
2. Substitute x = 4 and y = 12 into the derivative expression:
12 = e^(-a * 4)(b - a * 4)
Now, we have one equation with two variables (a and b). To solve for a and b, we need another piece of information.
If we have any additional information about the function, such as another point it passes through or its behavior at a specific point, it would help us solve for a and b. Without that information, we cannot determine the exact values of a and b.
However, if you know the values of a and b or have additional information, you can substitute them into the equation 12 = e^(-a * 4)(b - a * 4) to find the specific formula for the function.
y' = b(1-ax)e^(-ax)
1-ax=0, at x=1/a = 4, making a = 1/4
y(4) = 4be^(-4/4) = 12
4b/e = 12
b = 3e
y = 3ex e^(-x/4)
verify this at
https://www.wolframalpha.com/input/?i=y%3D3ex+e^%28-x%2F4%29