find the intervals of y=2x^2 - 19x+35
To find the intervals of the function y = 2x^2 - 19x + 35, we need to determine where the function is increasing or decreasing.
Step 1: Find the critical points by setting the derivative equal to zero and solving for x.
The derivative of y = 2x^2 - 19x + 35 is dy/dx = 4x - 19.
Setting this equal to zero, we have 4x - 19 = 0.
Solving for x, we get x = 19/4 = 4.75.
Step 2: Determine the intervals using the critical point.
Since the critical point is x = 4.75, we can use a number line to determine the intervals.
Choosing a number less than 4.75, such as x = 0, we can evaluate the sign of the derivative.
dy/dx = 4(0) - 19 = -19. Since this is negative, the function is decreasing in the interval (-∞, 4.75).
Choosing a number greater than 4.75, such as x = 5, we can evaluate the sign of the derivative.
dy/dx = 4(5) - 19 = 1. Since this is positive, the function is increasing in the interval (4.75, ∞).
Therefore, the intervals of the function y = 2x^2 - 19x + 35 are (-∞, 4.75) and (4.75, ∞).
To find the intervals of the quadratic function y = 2x^2 - 19x + 35, we need to determine where the function is increasing and decreasing.
Step 1: Find the critical points by taking the derivative of the function.
The derivative of y with respect to x is:
y' = 4x - 19
Set y' = 0 to find the critical points:
4x - 19 = 0
4x = 19
x = 19/4
Step 2: Test points in intervals to determine the sign of y'.
Choose test points within the intervals:
Interval 1: x < 19/4 (choose x = 0)
Interval 2: x > 19/4 (choose x = 5)
Step 3: Evaluate y' at the test points to determine the sign of y'.
Using the test points:
y'(0) = 4(0) - 19 = -19 (negative)
y'(5) = 4(5) - 19 = 1 (positive)
Step 4: Determine the intervals based on the sign of y':
From interval 1, y' is negative, which means the function is decreasing.
From interval 2, y' is positive, which means the function is increasing.
Interval 1: (-∞, 19/4)
Interval 2: (19/4, +∞)
Therefore, the quadratic function y = 2x^2 - 19x + 35 is decreasing on the interval (-∞, 19/4) and increasing on the interval (19/4, +∞).
To find the intervals where the function y = 2x^2 - 19x + 35 is positive or negative, we need to determine the x-values where the function crosses the x-axis or changes sign.
1) Start by setting the function equal to zero:
2x^2 - 19x + 35 = 0
2) Factorize the quadratic equation:
(2x - 5)(x - 7) = 0
3) Set each factor equal to zero and solve for x:
2x - 5 = 0
x - 7 = 0
Solving these equations gives:
x = 5/2 and x = 7
4) Now, plot these values on a number line:
------------------------5/2------7------------
5) Choose a test point within each interval to determine the sign of the function. For simplicity, let's choose x = 0 (a value to the left of 5/2) and x = 10 (a value to the right of 7).
Testing x = 0:
y = 2(0)^2 - 19(0) + 35
y = 35
Testing x = 10:
y = 2(10)^2 - 19(10) + 35
y = 165
From these tests, we can conclude the sign of the function in each interval:
- The function is positive between negative infinity and 5/2.
- The function is negative between 5/2 and 7.
- The function is positive between 7 and positive infinity.
Therefore, the intervals where y = 2x^2 - 19x + 35 is positive are (-∞, 5/2) and (7, +∞), while the interval where it is negative is (5/2, 7).