2 times a number that is at least 42
2 answers:
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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:
(3,-1)
Step-by-step explanation:
So we have the system of equations:
As directed, add straight down. The y-variable will cancel:
Now, divide both sides by 5. The left cancels:
So x is 3.
Now, substitute 3 for x in either of the equations:
Substitute 3 for x:
Multiply:
Subtract 9 from both sides:
So, our answer is: x=3, y=-1 or (3,-1).
And we are done :)
We have been given that you drop a ball from a window 50 metres above the ground. The ball bounces to 50% of its previous height with each bounce. We are asked to find the total distance traveled by up and down from the time it was dropped from the window until the 25th bounce.
We will use sum of geometric sequence formula to solve our given problem.
, where,
a = First term of sequence,
r = Common ratio,
n = Number of terms.
For our given problem ,
and
.
Therefore, the ball will travel 100 meters and option B is the correct choice.