f(x)=−8x^3+6ax^2−3bx+4 has a local minimum at x=1 and a local maximum at x=3. If a and b are the local minimum and maximum, respectively, what is the absolute value of a+b?
f(x) = −8x^3+6ax^2−3bx+4
f'(x) = -24x^2 + 12ax - 3b
Now, local extrema occur where f'(x) = 0.
Since we see that f(x) is quadratic, and we know two zeros, we know that those are the only two zeros.
So, we know that
f'(x) = k(x-1)(x-3)
= kx^2 - 4kx + 3k
Take the antiderivative to see that
f(x) = k/3 x^3 - 2kx^2 + 3kx + C
Equating coefficients with the given f(x), we have
k/3 = -8, so k = -24
-2k = 48 = 6a, so a = 8
3k = -72 = -3b, so b = 24
a+b = 32
Check:
f(x) = -8x^3 + 48x^2 - 72x + 4
f'(x) = -24x^2 + 96x - 72
= -24(x^2 - 4x + 3)
= -24(x-1)(x-3)
Thanks Steve.
But apparently, that's not the right answer, since a and b are not local minimum and maximum values.
Got me. f(x) has a and b as coefficients.
Don't see how they can also be the local min and max.
Maybe you can figure out what they're asking.
Hmmm.
f(x) = −8x^3+6ax^2−3bx+4
f(1) = -8+6a-3b+4 = -4+6a-3b
f(3) = -216+54a-9b+4 = -212+54a-9b
So, if a and b are min and max, then
-4+6a-3b=a
-212+54a-9b=b
or
5a-3b = 4
54a-10b = 212
a = 149/28
b = 211/28
a+b = 90/7
Check:
f(x) = -8x^3 + 447/14 x^2 - 633/28 x + 4
f(1) = 149/28
f(3) = 211/28
If that's not right, I'm stumped what they want.
It's not right either. I know, I'm stumped too.
But thanks for all the help.
i had trouble with this problem too
Could it be that the coefficients a and b are not the min and max a and b?
If f(x) = -8x^3 + 48x^2 - 72x + 4
Then f(1) = -28 and f(3) = 4
In that case, min+max (or, a+b) = -24
Oops. |a+b| = 24
To find the absolute value of a+b, we need to determine the values of a and b. Since f(x) has a local minimum at x=1 and a local maximum at x=3, we can use this information to find the values of a and b.
First, let's find the derivative of f(x) to obtain the critical points:
f'(x) = -24x^2 + 12ax - 3b.
To find the critical points, we set the derivative equal to zero:
-24x^2 + 12ax - 3b = 0.
Now, let's find the value of a by substituting x=1 into the equation:
-24(1)^2 + 12a(1) - 3b = 0,
-24 + 12a - 3b = 0. Equation (1)
Next, we'll find the value of b by substituting x=3 into the equation:
-24(3)^2 + 12a(3) - 3b = 0,
-216 + 36a - 3b = 0. Equation (2)
We now have a system of two equations with two variables (a and b). We can solve this system of equations to find the values of a and b.
Let's start by multiplying Equation (1) by 3 and Equation (2) by 12 to eliminate the b term:
-72 + 36a - 9b = 0,
-2592 + 432a - 36b = 0.
Next, we'll multiply Equation (1) by 4 and Equation (2) by 1 to eliminate the a term:
-96 + 48a - 12b = 0,
-216 + 36a - 3b = 0.
Adding these two equations together, we get:
-312 + 84a - 15b = 0.
Now we have a single equation in terms of a and b. Rearranging the equation, we find:
15b - 84a = 312. Equation (3)
Substituting Equation (1) into Equation (3), we can solve for b:
15b - 84(12) + 2592 = 312,
15b - 1008 + 2592 = 312,
15b + 1584 = 312,
15b = -1272,
b = -84.
Substituting the value of b into Equation (1), we can solve for a:
-24 + 12a - 3(-84) = 0,
-24 + 12a + 252 = 0,
12a = -228,
a = -19.
Therefore, the value of a is -19 and the value of b is -84. To find the absolute value of a+b, we calculate:
|a+b| = |-19 + (-84)| = |-103| = 103.
Hence, the absolute value of a+b is 103.