Given f(x)=x^2-27x-28, find the largest integer in shich f is decreasing.
f is decreasing when f' < 0
f' = 2x-27
so, when is 2x-27 < 0?
Negative infinity? Or 27/2?
To find the largest integer at which f(x) is decreasing, we need to determine the critical points of the function.
Step 1: Find the derivative of f(x)
The derivative of f(x) is obtained by applying the power rule of differentiation:
f'(x) = 2x - 27
Step 2: Set the derivative equal to zero to find critical points
To find the critical points of f(x), we set f'(x) = 0 and solve for x:
2x - 27 = 0
2x = 27
x = 27/2
Step 3: Determine the increasing and decreasing intervals
To determine if f(x) is increasing or decreasing in different intervals, we can check the sign of the derivative f'(x).
For x < 27/2:
We can choose a test point like x = 0 and evaluate the sign of f'(0):
f'(0) = 2(0) - 27 = -27
The derivative f'(x) is negative for x < 27/2, indicating that f(x) is decreasing in this interval.
For x > 27/2:
We can choose a test point like x = 10 and evaluate the sign of f'(10):
f'(10) = 2(10) - 27 = -7
The derivative f'(x) is negative for x > 27/2, indicating that f(x) is also decreasing in this interval.
Step 4: Find the largest integer for which f(x) is decreasing
Since the critical point x = 27/2 is not an integer, we need to find the largest integer below it. The largest integer below 27/2 is 13.
Therefore, the largest integer at which f(x) is decreasing is x = 13.
To find the largest integer in which the function f(x) = x^2 - 27x - 28 is decreasing, we need to analyze its derivative. The derivative of a function helps determine where the function is increasing or decreasing.
Step 1: Find the derivative of f(x)
Let's differentiate the function f(x) with respect to x:
f'(x) = 2x - 27
Step 2: Set the derivative equal to zero and solve for x
To find where f(x) is either increasing or decreasing, we set the derivative equal to zero:
2x - 27 = 0
Adding 27 to both sides of the equation, we get:
2x = 27
Dividing by 2, we find:
x = 27/2
Step 3: Determine the intervals of increase and decrease
To determine where the function f(x) is increasing or decreasing, we need to evaluate the derivative on different intervals.
- If f'(x) > 0, then f(x) is increasing
- If f'(x) < 0, then f(x) is decreasing
Let's choose test values from two different intervals:
- Test x = 0, which is in the interval (-∞, 27/2) [left side of 27/2]
- Test x = 10, which is in the interval (27/2, +∞) [right side of 27/2]
Substituting these values into the derivative:
- For x = 0, f'(0) = 2(0) - 27 = -27 (negative)
- For x = 10, f'(10) = 2(10) - 27 = -7 (negative)
Step 4: Determine the largest integer in which f(x) is decreasing
From the test values, we see that the derivative is negative for both x = 0 and x = 10. This means that the function f(x) is decreasing on the intervals (-∞, 27/2) and (27/2, +∞).
Since we want to find the largest integer in which f(x) is decreasing, we need to find the largest integer value that is less than or equal to 27/2 (but not including it). The largest integer smaller than 27/2 is 13.
Therefore, the largest integer in which f(x) is decreasing is 13.