6. let f and g be differentiable functions
with the following properties:
i. g(x) > 0 for all x
ii. f(0) = 1
if h(x) = f(x)g(x) and h'(x)
= f(x)g'(x), then what is f(x)?
(a) 0
(b) 1
(c) f'(x)
(d) e^x

With the given properties in the question, the option that represents the correct value of f(x) is;

Option B: 1

We are told that;

h(x) = f(x)g(x)

Thus;

h'(x) = (d/dx)[f(x)g(x)]

>> h'(x) = f(x)•g'(x) + g(x)•f'(x)

Now, we are given that;

h'(x) = f(x)g'(x)

Applying this property to our derivatives h'(x), we have;

f(x)g'(x) = f(x)•g'(x) + g(x)•f'(x)

Simplifying, f(x)g'(x) will cancel out to give us;

g(x)•f'(x) = 0

       Thus; f'(x) = 0

f(x) = d

Thus; f(0) = d

We are told that; f(0) = 1

By transitive property earlier proved, we can deduce that;

d = 1

Thus; f(x) = 1

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(b) 1

Step-by-step explanation:

To differentiate we will need the product rule:

.

We have , so the following equation is true by the transitive property:

By subtraction property we have:

Since , then we can divide both sides by :

This implies is constant.

So we have that where is a real number.

Since and , then by transitive property .

So .

Checking:

So the following conditions were met.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

a

step-by-step explanation: