Explanation:
The cubic ...
f(x) = ax³ +bx² +cx +d
has derivatives ...
f'(x) = 3ax² +2bx +c
f''(x) = 6ax +2b
<h3>a)</h3>
By definition, there will be a point of inflection where the second derivative is zero (changes sign). The second derivative is a linear equation in x, so can only have one zero. Since it is given that a≠0, we are assured that the line described by f''(x) will cross the x-axis at ...
f''(x) = 0 = 6ax +2b ⇒ x = -b/(3a)
The single point of inflection is at x = -b/(3a).
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<h3>b)</h3>
The cubic will have a local extreme where the first derivative is zero and the second derivative is not zero. These will only occur when the discriminant of the first derivative quadratic is positive. Their location can be found by applying the quadratic formula to the first derivative.

There will be zero or two local extremes. A local extreme cannot occur at the point of inflection, which is where the formula would tell you it is when there is only one.
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<h3>c)</h3>
Part A tells you the point of inflection is at x= -b/(3a).
Part B tells you the midpoint of the local extremes is x = -b/(3a). (This is half the sum of the x-values of the extreme points.) You will notice these are the same point.
The extreme points are located symmetrically about their midpoint, so are located symmetrically about the point of inflection.
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Additional comment
There are other interesting features of cubics with two local extremes. The points where the horizontal tangents meet the graph, together with the point of inflection, have equally-spaced x-coordinates. The point of inflection is the midpoint, both horizontally and vertically, between the local extreme points.
Answer:
The blu
Step-by-step explanation:
It is parallel, while the others are not.
6. scalene triangle
7. is a right angle
Step-by-step explanation:
Answer:
V= 8/3
Step-by-step explanation:
To solve this question, we need to understand sample and population concepts.
From this, the answer is:
The 25 females that Julie interviews are the sample, and all female college students at state universities are the population.
Example:
I want to estimate the proportion of New York state residents who are Buffalo Bills fans. So i ask, lets say, 1000 randomly selected New York state residents whether they are Buffalo Bills fans, and expand this to the entire population of New York State residents, which is the population of the study.
The sample will be: 1000 randomly selected New York state residents
The population will be: All New York state residents
In this question:
Sample: 25 females from each university in her state.
Population: All female college students in her student.
Thus, the correct statement is:
The 25 females that Julie interviews are the sample, and all female college students at state universities are the population.
For a similar example, you can check brainly.com/question/24331410