Find B so that F(x)=-x^2+bx-75 has a maximum value of 25.
the max is obtained at the vertex.
the x of the vertex is -b/-2 = b/2
so f(b/2) = 25
-b^2/4 + b^2/2 - 75 = 25
- b^2/4 + 2b^2/4 = 100
b^2/4 = 100
b^2 = 400
b = ± 20
Why are mathematicians never invited to parties? Because they always find the maximum value of the fun! But fear not, my friend, I shall help you find the mysterious constant B.
To find the value of B in the equation F(x) = -x^2 + bx - 75, where F(x) has a maximum value of 25, we need to remember that the maximum point lies on the vertex of the quadratic function.
In general, the x-coordinate of the vertex of a quadratic equation of the form F(x) = ax^2 + bx + c is given by x = -b/(2a). In this case, a = -1 (coefficient of x^2) and we want to find B (the coefficient of x), so b = B.
Plugging in the given maximum value of x = 25 into the equation x = -b/(2a), we have:
25 = -B/(2*-1)
25 = -B/(-2)
25 = B/2
To find the value of B, we can solve for it by multiplying both sides of the equation by 2:
25 * 2 = B
50 = B
So, the value of B that makes F(x) have a maximum value of 25 is B = 50. Voila!
To find the value of b, let's first differentiate the function F(x) with respect to x to find its critical points:
F(x) = -x^2 + bx - 75
Taking the derivative:
F'(x) = -2x + b
To find the critical points, we set the derivative equal to zero and solve for x:
-2x + b = 0
Rearranging the equation, we find:
-2x = -b
Dividing both sides by -2:
x = b/2
Now, let's substitute this value of x back into the original function to find the maximum value:
F(x) = -x^2 + bx - 75
Substituting x = b/2:
F(b/2) = -(b/2)^2 + b(b/2) - 75
Simplifying:
F(b/2) = -(b^2/4) + b^2/2 - 75
Multiplying both sides by 4 to get rid of the fraction:
4F(b/2) = -b^2 + 2b^2 - 300
Simplifying further:
4F(b/2) = b^2 - 300
Since we want the maximum value of 25, we set the expression equal to 25:
b^2 - 300 = 25
Rearranging the equation:
b^2 - 325 = 0
To solve for b, we can use the quadratic formula:
b = ±√(325)
So, there are two possible values for b:
b = √(325) or b = -√(325)
Thus, the possible values of b for which the function F(x) has a maximum value of 25 are b = √(325) or b = -√(325).
To find the value of B such that the function F(x) = -x^2 + bx - 75 has a maximum value of 25, we need to use a mathematical concept called "completing the square" and the properties of quadratic functions.
First, let's write the quadratic function F(x) in its vertex form:
F(x) = a(x - h)^2 + k
In this form, (h, k) represents the coordinates of the vertex of the parabola, and a determines the steepness of the graph.
In our case, F(x) = -x^2 + bx - 75. Comparing this equation with the vertex form, we can see that a = -1.
Now, let's complete the square to write F(x) in the vertex form:
F(x) = -x^2 + bx - 75
= -(x^2 - bx + 75)
= -(x^2 - bx + (b/2)^2) + (b/2)^2 - 75 (adding and subtracting (b/2)^2 to complete the square)
= -(x^2 - bx + (b/2)^2) + (b^2/4) - 75
Simplifying the equation, we get:
F(x) = -(x - (b/2))^2 + (b^2/4) - 75
From the equation, we can deduce that the maximum value of F(x) is obtained when (x - (b/2))^2 = 0 (since a negative value squared will always be zero), which means x = b/2.
Since we want the maximum value of F(x) to be 25, we can substitute x = b/2 and F(x) = 25 into the equation and solve for B:
25 = -((b/2) - (b/2))^2 + (b^2/4) - 75
25 = (b^2/4) - 75
Adding 75 to both sides:
100 = (b^2/4)
Multiplying both sides by 4:
400 = b^2
Taking the square root of both sides:
b = ±20
Therefore, there are two possible values of B - B = 20 or B = -20 - where the quadratic function F(x) has a maximum value of 25.