Find the critical numbers for f=5+10x-x^{2} in the interval [-3,3].Also find the minimum and maximum.
To find the critical numbers of a function f(x), we need to find the values of x where the derivative of f(x) is either zero or undefined. The critical numbers will give us potential minimum or maximum points on the graph of the function.
First, let's find the derivative of the given function f(x):
f(x) = 5 + 10x - x^2
f'(x) = d/dx(5 + 10x - x^2)
To differentiate this function, let's use the power rule:
f'(x) = 10 - 2x
Now, to find the critical numbers, we set the derivative equal to zero and solve for x:
10 - 2x = 0
2x = 10
x = 5/2
So, x = 5/2 is a critical number.
Next, we need to check the endpoints of the interval [-3, 3]. Let's evaluate the function at x = -3 and x = 3 to see if they are critical numbers as well:
f(-3) = 5 + 10(-3) - (-3)^2 = 5 - 30 - 9 = -34
f(3) = 5 + 10(3) - 3^2 = 5 + 30 - 9 = 26
Now, we have three potential critical numbers: x = -3, 5/2, and 3.
To find the minimum and maximum points, we need to compare the function values at these critical numbers and the endpoints.
- At x = -3, f(-3) = -34
- At x = 5/2, we can substitute it back into the original function: f(5/2) = 5 + 10(5/2) - (5/2)^2 = 5 + 25 - 25/4 = 125/4
- At x = 3, f(3) = 26
Therefore, the minimum and maximum values of the function f(x) = 5 + 10x - x^2 in the interval [-3, 3] are:
Minimum: f(-3) = -34
Maximum: f(5/2) = 125/4