Find the maximum or minimum value of function and the value of x when it occurs: -2x^2+5x+5.
done earlier, and you were correct.
but that wasn't the correct answer
The max occurs when x = -b/2a = 5/4, and is indeed 65/8
wolframalpha.com confirms this:
http://www.wolframalpha.com/input/?i=-2x^2%2B5x%2B5
To find the maximum or minimum value of a function, you need to calculate the vertex of the parabola. The vertex represents the highest or lowest point of the parabola, and the x-value of the vertex indicates where this value occurs.
The given function is -2x^2 + 5x + 5. To find the maximum or minimum value and the corresponding x-value, you can follow these steps:
Step 1: Convert the function to vertex form.
The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Let's rewrite the given function in vertex form:
f(x) = -2x^2 + 5x + 5
To convert it to vertex form, complete the square by following these steps:
1. Factor out the common factor (-2) from the first two terms:
f(x) = -2(x^2 - (5/2)x) + 5
2. Add and subtract the square of half the coefficient of x within the parentheses to maintain the equality. Half of (5/2) is (5/4).
f(x) = -2(x^2 - (5/2)x + (5/4) - (5/4)) + 5
3. Factor the quadratic within the parentheses:
f(x) = -2((x - (5/4))^2 - (25/16)) + 5
4. Simplify and multiply by -2:
f(x) = -2(x - (5/4))^2 + (25/8) + 5
Now the function is in vertex form. The vertex is at the point (h, k) = ((5/4), (25/8) + 5).
Step 2: Find the maximum or minimum value.
Since the coefficient of the x^2 term is -2, it is a negative value, indicating that the parabola opens downward. Therefore, the vertex represents the maximum value of the function.
The maximum value is given by the y-coordinate of the vertex. In this case, the maximum value is (25/8) + 5.
Step 3: Determine the x-value.
To find the x-value where the maximum (or minimum) occurs, simply use the x-coordinate of the vertex. In this case, the x-value is (5/4).
Therefore, the maximum value of the function is (25/8) + 5, and it occurs when x = (5/4).