Which set only includes rational numbers
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The given polynomial function has 1 relative minimum and 1 relative maximum.
<h3>What are the relative minimum and relative maximum?</h3>- The relative minimum is the point on the graph where the y-coordinate has the minimum value.
- The relative maximum is the point on the graph where the y-coordinate has the maximum value.
- To determine the maximum and the minimum values of a function, the given function is derivated(since the maximum or minimum is obtained at slope = 0)
The given function is
f(x) = 2x³ - 2x² + 1
derivating the above function,
f'(x) = 6x² - 4x
At slope = 0, f'(x) = 0 (for maximum and minimum values)
⇒ 6x² - 4x = 0
⇒ 2x(3x - 2) = 0
2x = 0 or 3x - 2 = 0
∴ x = 0 or x = 2/3
Then the y-coordinates are calculated by substituting these x values in the given function,
when x = 0;
f(0) = 2(0)³ - 2(0)² + 1 = 1
So, the point is (0, 1)
when x = 2/3;
f(2/3) = 2(2/3)³ - 2(2/3)² + 1 = 19/27
So, the point is (2/3, 19/27)
Since y = 1 is the largest value, the point (0, 1) is the relative maximum for the given function.
So, y = 19/27 is the smallest value, the point (2/3, 19/27) is the relative minimum for the given function.
Thus, option A is correct.
Learn more about the relative minimum and maximum here:
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Answer:
Part A:
Hence proved that Vector= ai + bj is perpendicular to the line ax + by = c.
Part B:
Slope of vector =
Step-by-step explanation:
Condition for perpendicular is:
Part A:
Consider the vector v = ai + bj
x component of vector=a
y component of vector=b
Slope of vector=
Consider the line ax + by = c:
Rearranging the equation:
ax+by=c
by=c-ax
y=
According to general equation of line:
Where m is the slope
In our case the slope of above line is:
According to the condition of perpendicular:
Hence proved that Vector= ai + bj is perpendicular to the line ax + by = c.
Part B:
Slope of vector is also calculated above.
Since the slope of vector is negative reciprocal of the slope of the given line:
According to equation of line ax + by = c
y=
According to general equation of line:
Where m is the slope
Slope of given line=m=
negative reciprocal of the slope of the given line =
Slope of vector =