Question

find an equation of the line that is tangent to the graph of the function f(x)=7/√x and parallel to the line 7x+2y-12=0

Answer 1

The line 7 x + 2 y -12= 0 can be written
\[
y = -\frac 7 2 x +6
\]
So it has a slope of \( -\frac 7 2 \)

Answer 2

First, find the slope of the line.
Then, take the derivative of the function.
\(f'(x) \) is the equation of the slope of \(f(x)\) for any value of x.
Since you need to know at what x value, the function has the same slope as the line, set the derivative of the function equal to the slope of the line and solve for x.
This value of x is the value of x of the function where the slope of the function is the same as the slope of the line.
Now use the value of x in the equation of the function and evaluate y.
This (x, y) ordered pair is the point on the function where the tangent parallel tot eh line intersects the function.
Now use this (x, y) point and the slope of the line to find the equation of the line you need.

Answer 3

\[
f(x)=\frac{7}{\sqrt{x}}\\
f'(x)= -\frac{7}{2 x^{3/2}}
\]

Answer 4

Now you can finish it as @mathstudent55 has suggested

Answer 5

Put the equation of the line as
\[
h(x)= -\frac 7 2 x +b
\]
I will explain to you how to find b. and we will be done.

Answer 6

At the point where the line is tangent, we must have
\[
-\frac{7}{2 x^{3/2}} = -\frac 7 2
\]
That gives that x=1

Answer 7

if x =1, then f(1) = 7 so h(1) =7 so
\[
h(1)= -\frac 7 2 +b=7\\
b= \frac {21}2\\
h(x)= -\frac 7 2 x + \frac {21}2
\]