Given that the graph of f(x) passes through the point ( 9, 7 ) and that the slope of its tangent line at (x,f(x)) is 3 x + 5, what is f( 2 )?

To find the value of f(2), we need to use the given information and find the equation of the tangent line at the point (9, 7).

We know that the slope of the tangent line, at any point (x, f(x)), is given by 3x + 5. We can use this information to find the equation of the tangent line.

The equation of a straight line can be written in the slope-intercept form, y = mx + b, where m is the slope of the line and b is the y-intercept.

Since the point (9, 7) lies on the line, we can substitute the values x = 9 and y = 7 into the equation to find the value of b.

7 = (3 * 9) + b
7 = 27 + b
b = -20

Now we have the equation of the tangent line as y = (3x + 5)x - 20.

To find f(2), we substitute x = 2 into the equation:

f(2) = (3 * 2 + 5) * 2 - 20
f(2) = (6 + 5) * 2 - 20
f(2) = 11 * 2 - 20
f(2) = 22 - 20
f(2) = 2

Therefore, f(2) = 2.

To find f(2), we need to determine the equation of the graph of f(x) using the given information.

We are given that the graph of f(x) passes through the point (9, 7) and that the slope of its tangent line at (x, f(x)) is 3x + 5.

Since the slope of the tangent line represents the derivative of f(x), we can find the derivative of f(x) by taking the derivative with respect to x of the equation 3x + 5.

The derivative of 3x + 5 is simply 3.

Now, we need to find the equation of f(x) using the derivative and the given point (9, 7). We'll use the point-slope form of a linear equation.

The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

For our equation, we have:
y - 7 = 3(x - 9)

Expanding and simplifying, we get:
y - 7 = 3x - 27

Rearranging the equation, we find:
y = 3x - 20

So, the equation of the graph of f(x) is y = 3x - 20.

To find f(2), we substitute x = 2 into the equation:
f(2) = 3(2) - 20

Simplifying, we get:
f(2) = 6 - 20
f(2) = -14

Therefore, f(2) is -14.

f ' (x) = 3x+5

f(x) = (3/2)x^2 + 5x + c
but (9,7) lies on it, so
7 = (3/2)(81) + 5(9) + c
c = -319/2

f(x) = (3/2)x^2 + 5x - 319/2
f(2) = (3/2)(4) + 5(2) - 319/2