09:30 to 17:00 minus 30 minutes How many hours is that ?
1 answer:
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Answer: (a) Interval where f is increasing: (0.78,+∞);
Interval where f is decreasing: (0,0.78);
(b) Local minimum: (0.78, - 0.09)
(c) Inflection point: (0.56,-0.06)
Interval concave up: (0.56,+∞)
Interval concave down: (0,0.56)
Step-by-step explanation:
(a) To determine the interval where function f is increasing or decreasing, first derive the function:
f'(x) = [
]
Using the product rule of derivative, which is: [u(x).v(x)]' = u'(x)v(x) + u(x).v'(x),
you have:
f'(x) =
f'(x) =
f'(x) =
Now, find the critical points: f'(x) = 0
= 0
x = 0
and
x =
x = 0.78
To determine the interval where f(x) is positive (increasing) or negative (decreasing), evaluate the function at each interval:
interval x-value f'(x) result
0<x<0.78 0.5 f'(0.5) = -0.22 decreasing
x>0.78 1 f'(1) = 1 increasing
With the table, it can be concluded that in the interval (0,0.78) the function is decreasing while in the interval (0.78, +∞), f is increasing.
Note: As it is a natural logarithm function, there are no negative x-values.
(b) A extremum point (maximum or minimum) is found where f is defined and f' changes signs. In this case:
- Between 0 and 0.78, the function decreases and at point and it is defined at point 0.78;
- After 0.78, it increase (has a change of sign) and f is also defined;
Then, x=0.78 is a point of minimum and its y-value is:
f(x) =
f(0.78) =
f(0.78) = - 0.092
The point of <u>minimum</u> is (0.78, - 0.092)
(c) To determine the inflection point (IP), calculate the second derivative of the function and solve for x:
f"(x) = [
]
f"(x) =
f"(x) =
= 0
and
Substituing x in the function:
f(x) =
f(0.56) =
f(0.56) = - 0.06
The <u>inflection point</u> will be: (0.56, - 0.06)
In a function, the concave is down when f"(x) < 0 and up when f"(x) > 0, adn knowing that the critical points for that derivative are 0 and 0.56:
f"(x) =
f"(0.1) =
f"(0.1) = - 0.21, i.e. <u>Concave</u> is <u>DOWN.</u>
f"(0.7) =
f"(0.7) = + 1.33, i.e. <u>Concave</u> is <u>UP.</u>
Answer:
No, (false). The question I assume was:
Step-by-step explanation:
1. Multiple: 3 * 8 = 24
2. Multiple: <em><u>The result of step No. 1:</u></em> * 1 = 24 * 1 = 24
3. Conversion a decimal number to a fraction: 8.1 = =
a) Write down the decimal 8.1 divided by 1: 8.1 =
b) Multiply both top and bottom by 10 for every number after the
decimal point. (For example, if there are two numbers after the decimal
point, then use 100, if there are three then use 1000, etc.)
=
Note: is called a decimal fraction.
c) Simplify and reduce the fraction
4. Multiple: 3 * 8.1 = =
Multiply both numerators and denominators. Result fraction keep to lowest
possible denominator GCD(243, 10) = 1. In the next intermediate step the
fraction result cannot be further simplified by canceling.
In words - three multiplied by eighty-one tenths = two hundred forty-three
tenths.
5. Compare: <em><u>The result of step No. 2 </u></em>vs <em><u>The result of step No. 4</u></em>: = 24 vs =
vs
=
vs
=
vs
= 240 vs 243 = No