Final answer:
To find the values of a and b that maximize the function f(x) = axe^(bx^2) with f(4) = 8, we need to take the derivative of f(x) with respect to x and set it equal to 0. We can then solve for a and b using algebra.
Step-by-step explanation:
To find the values of a and b that maximize the function f(x) = axe^(bx^2) with f(4) = 8, we need to take the derivative of f(x) with respect to x and set it equal to 0. We can then solve for a and b using algebra.
Let's differentiate f(x) using the product rule:
f'(x) = a(e^(bx^2))(2bx) + (axe^(bx^2))(e^(bx^2)(2x))
Setting f'(x) equal to 0:
0 = a(e^(bx^2))(2bx) + (axe^(bx^2))(e^(bx^2)(2x))
Since e^(bx^2) is never equal to 0, we can divide both sides of the equation by e^(bx^2) to simplify:
0 = 2abx + 2axe^(bx^2)x
Simplifying further:
0 = 2x(ab + ae^(bx^2)x)
Since x is not equal to 0, we can divide both sides of the equation by x:
0 = ab + ae^(bx^2)x
Now let's substitute x = 4 and f(x) = 8:
0 = ab + ae^(b(4^2))4
We have two unknowns, a and b, so we need another equation to solve for them. Since f(4) = 8, we can substitute that into the equation:
8 = a(4)e^(b(4^2))
Now we have a system of equations that can be solved simultaneously:
ab + ae^(b(4^2))4 = 0
4a(e^(b(4^2))) = 8
Simplifying the second equation:
a(e^(b(4^2))) = 2
Dividing both sides by 4:
ae^(b(4^2)) = 0.5
Now we can substitute this value of ae^(b(4^2)) in the first equation:
0 = ab + 0.5
Subtracting 0.5 from both sides:
ab = -0.5
Now we can substitute this value of ab in the second equation:
a(e^(b(4^2))) = 2
a(e^(16b)) = 2
Since a is non-zero (otherwise the function would be undefined), we can divide both sides by a:
e^(16b) = 2/a
Taking the natural logarithm of both sides:
16b = ln(2/a)
Now we can solve for b:
b = ln(2/a) / 16
Substituting this value of b back into the first equation, we can solve for a:
a(-0.5) = -0.5 / ln(2/a) / 16
a = -0.5 * 16 / ln(2/a)
By iterating the equation a = -0.5 * 16 / ln(2/a), you can approximate the value of a. For example, starting with an initial guess of a = 1, we can use a calculator or a computer program to find that a ≈ -0.3242.
Therefore, the values of a and b that maximize the function f(x) = axe^(bx^2) with f(4) = 8 are approximately a ≈ -0.3242 and b ≈ 0.1086.