The bill for car repairs is $425 for parts and labor. The cost of parts is $226 and labor is $36.00 per hour. If h represents th
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Answer:
(a) The function is increasing and decreasing
(b) The local minimum is x = 5 and the maximum is x = -6
(c) The inflection point is
(d) The function is concave upward on and concave downward on
Step-by-step explanation:
(a) To find the intervals where is increasing or decreasing you must:
1. Differentiate the function
2. Now we want to find the intervals where f'(x) is positive or negative. This is done using critical points, which are the points where f'(x) is either 0 or undefined.
These points divide the number line into three intervals:
,
, and
Evaluate f'(x) at each interval to see if it's positive or negative on that interval.
Therefore f(x) is increasing and decreasing
(b) Now that we know the intervals where f(x) increases or decreases, we can find its extremum points. An extremum point would be a point where f(x) is defined and f'(x) changes signs.
We know that:
- f(x) increases before x = -6, decreases after it, and is defined at x = -6. So f(x) has a relative maximum point at x = -6.
- f(x) decreases before x = 5, increases after it, and is defined at x = 5. So f(x) has a relative minimum point at x = 5.
(c)-(d) An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa).
Concave upward is when the slope increases and concave downward is when the slope decreases.
To find the inflection points of f(x), we need to use the f''(x)
We set f''(x) = 0
Analyzing concavity, we get
The function is concave upward on because the f''(x) > 0 and concave downward on
because the f''(x) < 0.
f(x) is concave down before , concave up after it. So f(x) has an inflection point at
.
Answer:
The sum of the first 47 terms of the series, 12, 16, 20, ... S₄₈ is 4,888
Step-by-step explanation:
The given series is;
12, 16, 20, ...,
Therefore, the first term of the series is, a = 12
The common difference of series is found as follows;
The difference between subsequent terms, 12 and 16 is 16 - 12 = 4
The difference between subsequent terms, 16, and 20 is 20 - 16 = 4
Therefore, the common difference, d = 4
The series is therefore an arithmetic projection, AP
The sum of the first 'n' terms of an AP, Sₙ, is given as follows;
(47/2)*(2*12+(47-1)*4)
The sum of the first 47 terms is therefore given as follows;
The sum of the first 47 terms of the series, 12, 16, 20, ... S₄₈ = 4,888