Find constants a and b in the function f(x)=axe^(bx)such that f(1/7)=1 and the function has a local maximum at x=1/7.

a =
b =

a= 7e

b= -7

To find the constants a and b in the function f(x) = axe^(bx), we can use the given information that f(1/7) = 1 and the function has a local maximum at x = 1/7.

First, let's calculate the value of f(1/7) using the given information:

f(1/7) = a * (1/7) * e^((1/7) * b)

Since f(1/7) = 1, we can set up the equation:

1 = a * (1/7) * e^((1/7) * b)

Next, let's find the derivative of f(x) with respect to x:

f'(x) = a * e^(b * x) + a * x * e^(b * x) * b

To find the local maximum at x = 1/7, we can find the derivative f'(x) and set it equal to 0:

0 = a * e^((1/7) * b) + a * (1/7) * e^((1/7) * b) * b

Now we have a system of two equations:

1 = a * (1/7) * e^((1/7) * b)
0 = a * e^((1/7) * b) + a * (1/7) * e^((1/7) * b) * b

Simplifying the second equation gives:

0 = a * e^((1/7) * b) * (1 + (1/7) * b)

Dividing the second equation by the first equation gives:

0 = (1 + (1/7) * b) / (1/7)

Simplifying further gives:

0 = 7 + b

So, b = -7.

Substituting the value of b into the first equation gives:

1 = a * (1/7) * e^((1/7) * (-7))
1 = a * (1/7) * e^(-1)
1 = a * (1/7) * 1/e

Multiply both sides by 7:

7 = a / e

Multiply both sides by e to solve for a:

a = 7e

Therefore, the constants a and b in the function f(x) = axe^(bx) such that f(1/7) = 1 and the function has a local maximum at x = 1/7 are:

a = 7e
b = -7

To find the constants a and b in the function f(x) = axe^(bx) such that f(1/7) = 1 and the function has a local maximum at x = 1/7, we can use the properties of local maximum.

First, let's find the value of a.

Since f(1/7) = 1, we can substitute x = 1/7 into the function:

f(1/7) = a(1/7)e^((1/7)b) = 1

Now, let's find the value of b using the fact that f(x) has a local maximum at x = 1/7.

For a function to have a local maximum, its first derivative must be equal to 0 at that point. Hence, we'll find the first derivative of f(x) and set it equal to 0:

f'(x) = a(e^(bx) + bxe^(bx)) = 0

Substituting x = 1/7 into the above equation, we get:

f'(1/7) = a(e^(b/7) + (b/7)e^(b/7)) = 0

To satisfy this equation, either a or (e^(b/7) + (b/7)e^(b/7)) must be equal to 0. However, since f(1/7) = 1, a cannot be equal to 0. Therefore, we must have:

e^(b/7) + (b/7)e^(b/7) = 0

Solving this equation for b analytically is challenging, so we can use numerical methods or approximation techniques to find an approximate value for b.

Hence, the values of a and b cannot be determined exactly with the given information.