Solution
David answered on
Dec 22 2021
Sol: (3) (a) 2 ,1
x
f x
x
(b)
3
4
,
1
x
g x
x
(c)
3
6
,
1
x x
h x
x
(d)
3
4
,
1
x
k x
x
Sol: (2)
The given function
2 on ,0f x x
First we evaluate the local extreme. Take the first derivative of the given function,
' 1f x
But there is no place where ' 0f x , so we don’t have any local extreme to be
concerned about.
That means we only evaluate at the end points.
0 2
f
f
So the absolute maximum is 2.
And the absolute minimum is .
Hence the function has only absolute extreme values.
Sol: (20)
The given function
2
2 4 ,f x x
First we evaluate the local extreme. Take the first derivative of the given function,
2' 2 2 4f x x x
To find the local extreme values, ' 0f x
2
2
2 2 4 0
2 4 0
0,2, 2
x x
x x
x
Now obtain the function values,
At 0,x
2
2
2
4
0 0 4
0 16
f x x
f
f
At 2,x
2
2 4 4
2 0
f
f
At 2,x
2
2 4 4
2 0
f
f
Sol: (34)
The given function
22 ,xf x x e
First we evaluate the local extreme. Take the first derivative of the given function,
2 2
2
2
2
' 2 2
' 2 1
x x
x
f x x e x e x
f x xe x
To find the local extreme values, ' 0f x
2
2
2
2
2 1 0
1 0
1,0,1
x
x
xe x
xe x
x
Now obtain the extreme values,
At 1,x
2
2
2
2 1
1
1 1
1 0.368
xf x x e
f e
f e
At 0,x
22
0 0
xf x x e
f
At 1,x
2
2
2
2 1
1
1 1
1 0.368
xf x x e
f e
f e
So these are all the local extreme values of the given function.
Sol: (6)
The given function is
3 510 3f x x x
On taking a first derivative of f x ,
2 4' 30 15f x x x
On taking a second derivative of f x ,
...