Answer:
(a) The critical points of f are x=0 and x=3.
(b)f is decreasing on
and f is decreasing on
.
(c) Therefore the local minimum of f is at x=3
Step-by-step explanation:
Given function is
![f'(x)= x^{-\frac35}(x-3)](https://tex.z-dn.net/?f=f%27%28x%29%3D%20x%5E%7B-%5Cfrac35%7D%28x-3%29)
(a)
To find the critical point set f'(x)=0
![\therefore x^{-\frac35}(x-3)=0](https://tex.z-dn.net/?f=%5Ctherefore%20%20x%5E%7B-%5Cfrac35%7D%28x-3%29%3D0)
![\Rightarrow x=0,3](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D0%2C3)
The critical points of f are 0,3.
(b)
The interval are
and
.
To find the increasing or decreasing, taking two points one point from the interval (0,3) and another point
.
Assume 1 and 4.
Now ![f'(1)=(1)^{-\frac35}(1-3)](https://tex.z-dn.net/?f=f%27%281%29%3D%281%29%5E%7B-%5Cfrac35%7D%281-3%29%3C0)
and ![f'(4)=(4)^{-\frac35}(4-3)>0](https://tex.z-dn.net/?f=f%27%284%29%3D%284%29%5E%7B-%5Cfrac35%7D%284-3%29%3E0)
Since 1∈
, f'(x)<0 and 4∈
, f'(x)>0
∴f is decreasing on
and f is decreasing on
.
(c)
![f'(x)= x^{-\frac35}(x-3)](https://tex.z-dn.net/?f=f%27%28x%29%3D%20x%5E%7B-%5Cfrac35%7D%28x-3%29)
Differentiating with respect to x
![f''(x)=-\frac35x^{-\frac 85}(x-3)+x^{-\frac35}](https://tex.z-dn.net/?f=f%27%27%28x%29%3D-%5Cfrac35x%5E%7B-%5Cfrac%2085%7D%28x-3%29%2Bx%5E%7B-%5Cfrac35%7D)
Now
![f''(0)=-\frac35(0)^{-\frac 85}(0-3)+(0)^{-\frac35}=0](https://tex.z-dn.net/?f=f%27%27%280%29%3D-%5Cfrac35%280%29%5E%7B-%5Cfrac%2085%7D%280-3%29%2B%280%29%5E%7B-%5Cfrac35%7D%3D0)
and
![f''(3)=-\frac35(3)^{-\frac 85}(3-3)+3^{-\frac35}](https://tex.z-dn.net/?f=f%27%27%283%29%3D-%5Cfrac35%283%29%5E%7B-%5Cfrac%2085%7D%283-3%29%2B3%5E%7B-%5Cfrac35%7D)
![=0.517>0](https://tex.z-dn.net/?f=%3D0.517%3E0)
Since f''(x)>0 at x=3
Therefore the local minimum of f is at x=3
Answer:
57°F differences
Step-by-step explanation:
-14+57= 43
43-57= -14
Answer:
b = 3/4
Step-by-step explanation:
The area of a triangle is given by
A = 1/2 bh
3/ 40 = 1/2 b ( 1/5)
Multiply
3/40 = 1/10 b
Multiply each side by 10
10 *3/40 = 10* 1/10 b
3/4 = b
07:10 = 7:10 AM
17:30 = 5:30 PM