Find the values of b such that the function has the given maximum value.
f(x) = −x2 + bx − 11;
Maximum value: 53
the vertex of a parabola is at the point
(-b/2a , c - b^2/4a)
so, you want
-11 + b^2/4 = 53
so solve for b. Check to be sure you get the right solution.
F(x) = -x^2 + Bx - 11.
Xv = h =-B/-2 = B/2.
-(B/2)^2 + B*B/2 - 11 = 53.
-B^2/4 + B^2/2 - 11 = 53,
-B^2/4 + 2B^2/4 - 11 = 53,
B^2/4 = 64,
B^2 = 256,
B = 16.
Xv = h = B/2 = 16/2 = 8.
V(h, k) = V(8, 53).
Well, finding the values of b such that the function has a maximum value of 53 is definitely no joke! But don't worry, I'm here to make it a little more amusing for you.
To find the values of b, we need to remember that the vertex of the parabola represents the maximum value of the function. And lucky for us, the formula for the x-coordinate of the vertex is given by:
x = -b / (2a)
In our case, a is equal to -1 (the coefficient of x^2), so we can rewrite the formula as:
x = b / 2
Since we know that the maximum value occurs at x, we can substitute this into our function:
f(x) = -(x^2) + bx - 11
f(x) = -(b^2 / 4) + (b^2 / 2) - 11
Simplifying this equation, we get:
f(x) = -(b^2 / 4) + (2b^2 / 4) - 11
f(x) = (3b^2 - 44) / 4
Now, since we want the maximum value to be 53, we can set f(x) equal to 53:
(3b^2 - 44) / 4 = 53
To get rid of the fraction, we can multiply both sides of the equation by 4:
3b^2 - 44 = 212
Let's continue with some math magic:
3b^2 = 256
b^2 = 256 / 3
b = ±sqrt(256 / 3)
Voilà! The values of b that make the function have a maximum value of 53 are approximately b ≈ ±6.84. Feel free to have a laugh while you give it a try on your own!
To find the values of b such that the function f(x) has a maximum value of 53, we can use the concept of the vertex of a quadratic function.
The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.
In this case, the quadratic function is f(x) = -x^2 + bx - 11, and the maximum value is 53.
To find the values of b, we need to determine the x-coordinate of the vertex of the function. The x-coordinate of the vertex is given by -b/2a.
Using the given quadratic function, we have a = -1, b = b, and c = -11.
Plugging these values into the formula for the x-coordinate of the vertex, we get:
x-coordinate of the vertex = -b / (2 * -1)
= b / 2
Now, to find the maximum value of the function, we substitute the x-coordinate of the vertex into the function:
f(x-coordinate of the vertex) = -((b / 2)^2) + b * (b / 2) - 11
= -(b^2 / 4) + (b^2 / 2) - 11
= (b^2 / 2) - (b^2 / 4) - 11
= (2b^2 - b^2) / 4 - 11
= b^2 / 4 - 11
Since the maximum value of the function is given as 53, we have:
b^2 / 4 - 11 = 53
To solve this equation, we can isolate b^2 on one side:
b^2 / 4 = 53 + 11
b^2 / 4 = 64
Next, we can multiply both sides of the equation by 4 to get rid of the fraction:
b^2 = 4 * 64
b^2 = 256
Finally, we take the square root of both sides to solve for b:
b = ±√256
b = ±16
Therefore, the values of b that make the function have a maximum value of 53 are b = 16 and b = -16.
To find the values of b such that the function f(x) = -x^2 + bx - 11 has a maximum value of 53, we need to use the concept of completing the square. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Step 1: Rewrite the quadratic function in vertex form:
f(x) = -(x^2 - bx + 11)
= -(x^2 - bx + (b/2)^2 - (b/2)^2 + 11)
Step 2: Simplify the expression inside the parentheses by completing the square:
f(x) = -(x^2 - bx + (b/2)^2) + (b/2)^2 - 11
Step 3: Rearrange the terms:
f(x) = -(x - b/2)^2 + (b^2/4) - 11
From the vertex form, we can determine the vertex by identifying the values of h and k:
- The x-coordinate of the vertex is given by h = b/2.
- The y-coordinate of the vertex is given by k = (b^2/4) - 11.
Since the maximum value of the function is 53, we equate k to 53:
53 = (b^2/4) - 11
Step 4: Solve for b:
(b^2/4) = 53 + 11
b^2 = 4(53 + 11)
b^2 = 4(64)
b^2 = 256
b = ± √256
b = ± 16
Therefore, the values of b such that the function f(x) = -x^2 + bx - 11 has a maximum value of 53 are b = 16 and b = -16.