Find the maximum value of 5x + 7 – x2 on the horizontal span of 0 to 5. (Enter the maximum value of the expression, not the value of x at that point.)
To find the maximum value of the expression 5x + 7 - x^2 on the horizontal span of 0 to 5, we can follow these steps:
Step 1: Calculate the derivative of the expression with respect to x.
Let's differentiate the expression 5x + 7 - x^2 with respect to x. The derivative will help us identify the critical points where the maximum or minimum occurs.
The derivative of 5x with respect to x is 5, and the derivative of -x^2 with respect to x is -2x. So, the derivative of the entire expression is:
d/dx (5x + 7 - x^2) = 5 - 2x
Step 2: Set the derivative equal to zero and solve for x.
To find the critical points, we set the derivative 5 - 2x equal to zero and solve the equation:
5 - 2x = 0
Solving for x, we get:
-2x = -5
x = 5/2
Step 3: Evaluate the expression at the critical points and endpoints.
We need to evaluate the expression 5x + 7 - x^2 at each of the critical points and the endpoints of the given horizontal span, which is from x = 0 to x = 5.
Let's substitute the critical points (x = 5/2) and the endpoints (x = 0 and x = 5) into the expression:
At x = 0: 5(0) + 7 - (0)^2 = 7
At x = 5/2: 5(5/2) + 7 - (5/2)^2 = 21/2
At x = 5: 5(5) + 7 - (5)^2 = 7
Step 4: Identify the maximum value.
Comparing the values obtained above, we see that the maximum value occurs at x = 5/2, where the value of the expression is 21/2.
Therefore, the maximum value of the expression 5x + 7 - x^2 on the horizontal span of 0 to 5 is 21/2.