find the values a and b must have to make the function f(x) = axebx have the following properties, rather than the ones the book requires:
Make f([ 5/4]) = 2, and
make sure f has a local maximum when x = [ 5/4].
a =
b =
To find the values of a and b that satisfy the given properties for the function f(x) = axebx, we need to use the information f([5/4]) = 2 and f has a local maximum when x = [5/4].
Step 1: Substituting x = [5/4] into the function, we have:
f([5/4]) = a[5/4]eb[5/4] = 2
Step 2: To find the values of a and b, we need to eliminate one of the variables. We can choose to eliminate b by taking the natural logarithm of both sides of the equation:
ln(f([5/4])) = ln(2)
ln(a[5/4]eb[5/4]) = ln(2)
ln(a) + ln([5/4]) + [5/4]ln(b) = ln(2)
Step 3: Now we can differentiate the function f(x) with respect to x and find the critical points where the derivative equals zero. Since we want f(x) to have a local maximum when x = [5/4], the derivative must equal zero at x = [5/4].
Differentiating f(x) = axebx with respect to x, we have:
f'(x) = aebx + axebx(b)
Step 4: Setting the derivative equal to zero and solving for b, we get:
aebx + axebx(b) = 0
aebx(1 + bx) = 0
This equation will only be true if either a = 0 or (1 + bx) = 0. However, we want to find values of a and b that satisfy the given properties, so a cannot be zero.
Therefore, we need to set (1 + bx) = 0 to find a non-zero value for b:
1 + bx = 0
bx = -1
b = -1/x
Step 5: Substitute the value of b into the equation we obtained in Step 2:
ln(a) + ln([5/4]) + [5/4]ln(-1/x) = ln(2)
Step 6: Since we have an equation with the variables a and x, we can rearrange the equation and solve for a:
ln(a) = ln(2) - ln([5/4]) - [5/4]ln(-1/x)
a = e^(ln(2) - ln([5/4]) - [5/4]ln(-1/x))
Now, with the value of b and a, the function f(x) = axebx will have the desired properties.
To find the values of a and b that satisfy the given requirements for the function f(x) = axebx, we need to consider both properties separately.
1. Making f([5/4]) = 2:
To satisfy this condition, we substitute x = 5/4 into the function and set it equal to 2:
f([5/4]) = a(5/4)e(5/4)b = 2
To determine the values of a and b, we need to know the value of e (the base of natural logarithm) and solve for a and b simultaneously.
2. Making f have a local maximum when x = [5/4]:
For f to have a local maximum at x = [5/4], its derivative should be zero at that point. Let's find the derivative of f(x) with respect to x, and then set it equal to zero:
f'(x) = (d/dx)[axebx]
= abxebx + aebx(d/dx)[bx]
= abxebx + aebx(b)
= abxebx + ab2xebx
Now we substitute x = 5/4 into the derivative, and set it equal to zero:
0 = ab(5/4)e(5/4)b + ab2(5/4)e(5/4)b
Solving equations 1 and 2 will give us the values of a and b that satisfy the given conditions.
Please provide the value of e (the base of natural logarithm) to continue.