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.