(a) When f is increasing the derivative of f is positive.
f'(x) = 15x^4 - 15x^2 > 0
15x^2(x^2 - 1)> 0
x^2 - 1 > 0 (The inequality doesn't flip sign since x^2 is positive)
x^2 > 1
Then f is increasing when x < -1 and x > 1.
(b) The f is concave upward when f''(x) > 0.
f''(x) = 60x^3 - 30x > 0
30x(2x^2 - 1) > 0
x(2x^2 - 1) > 0
x(x^2 - 1/2) > 0
x(x - 1/sqrt(2))(x + 1/sqrt(2)) > 0
There are four regions here. We will check if f''(x) > 0.
x < -1/sqrt(2): f''(-1) = -30 < 0
-1/sqrt(2) < x < 0: f''(-0.5) = 7.5 > 0
0 < x < 1/sqrt(2): f''(0.5) = -7.5 < 0
x > 1/sqrt(2): f''(1) = 30 > 0
Thus, f''(x) > 0 at -1/sqrt(2) < x < 0 and x > 1/sqrt(2).
Therefore, f is concave upward at -1/sqrt(2) < x < 0 and x > 1/sqrt(2).
(c) The horizontal tangents of f are at the points where f'(x) = 0
15x^2(x^2 - 1) = 0
x^2 = 1
x = -1 or x = 1
f(-1) = 3(-1)^5 - 5(-1)^3 + 2 = 4
f(1) = 3(1)^5 - 5(1)^3 + 2 = 0
Therefore, the tangent lines are y = 4 and y = 0.
Eliminationg these equations gives zero ,no solution
A.P. series would be: 201, 204, 207 .... 597
Here, a = 201, d = 3, l = 597
Now, l = a + (n - 1) d
597 = 201 + (n - 1)3
597 - 201 = 3n - 3
396 + 3 = 3n
n = 399/3
n = 133
In short, Your Answer would be 133
Hope this helps!
Answer:
5:12
Step-by-step explanation:
125:300 simplified = 5:12
I hope this helps
Answer:
Growth rate is 17%
2025 value is $76
Step-by-step explanation:
Here, we want to determine the annual growth rate
The general equation form is;
F = l( 1 + r)^n
where F is the future value
I is the initial value
r is the growth rate
n is the numbering years
To get the growth rate, we simply equate what we have in the bracket to 1 + r
Thus;
1 + r = 1.17
r = 1.17-1
r = 0.17
0.17 is same as 17/100 which is same as 17%
Secondly, we want to get the value of the book in 2025
What we have to do here is to substitute (2025-2000) for t which is 25
We have;
V(t) = 1.50(1.17)^25
V(25) = 75.986 which is approximately 76
