Consider the quadratic function
f(x) = – x^2 + 10x – 26. Determine whether there is a maximum or
minimum value and find that value.
This is algebra?
If you have not learned differentiation, perhaps your teacher wants you to use a calculator to find the extrema. (If you have, disregard this paragraph).
Take the derivative of f(x):
f'(x)= -2x + 10
Set f'(x) = 0
-2x + 10 = 0
10 = 2x
x = 5
Now use a sign line to find whether x=5 is a minimum or maximum.
f'(0) = +
f'(10) = -
x=5 is a maximum because f'(x) changes signs from + to -
complete the square,
f(x) = – x^2 + 10x – 26
= - [x^2 - 10x + 25 - 25] - 26
= -(x-5)^2 + 25 - 26
= -(x-5)^2 - 1
so the vertex is (5,-1) and since the parabola opens downwards, it will be a maximum point and the maximum value of the function is -1
I do not think these answers are right. I get somethin else and 5 is not an option. Here are the choices
A. Minimum is 25
B. Minimum is -51
C. Maximum is -1
D. Maximum us -51
5 is the value at which the maximum occurs. f(5) = -1, so choice C.
I was thinking that as well! Thanks!
that is exactly the answer I gave you, read my last line of my reply please
To determine whether there is a maximum or minimum value for the quadratic function, we need to consider its graph, which is a parabola.
The general form of a quadratic function is f(x) = ax^2 + bx + c. In our case, f(x) = -x^2 + 10x - 26.
The coefficient of the x^2 term, a, is negative (-1 in this case). When the coefficient a is negative, the parabola opens downwards, indicating that the function has a maximum value.
To find the maximum value, we can use the vertex formula. The vertex formula states that for a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex (h) can be found using the formula h = -b / (2a).
In our case, a = -1 and b = 10. Plugging in these values into the formula, we get h = -(10) / (2(-1)) = -10 / -2 = 5.
So the x-coordinate of the vertex is 5. To find the corresponding y-coordinate, we substitute this x-value into the quadratic function:
f(5) = -5^2 + 10(5) - 26 = -25 + 50 - 26 = -1.
Therefore, the maximum value of the quadratic function f(x) = -x^2 + 10x - 26 is -1, and it occurs at x = 5.
Hence, the function has a maximum value of -1 at x = 5.