The function f(x)=x 3 +1 4 x−1 4 is a monotonically increasing function, hence it is injective (one-to-one), so its inverse function exists and is well defined. How many points of intersection are there, between the function f(x) and its inverse f −1 (x) ?

I assume you meant:

f(x) = x^3 + 14x - 14
(I don't understand why you would leave spaces between the digits)

so y = x^3 +14x - 14
its inverse is x = y^3 + 14y - 14

to intersect them, solve the two equations.
so
y = (y^3 + 14y - 14)^3 + 14(y^3 + 14y - 14) - 14
that looks like a nasty mess to solve, BUT

I noticed in the original, if x = 1 , I get y = 1
and in the inverse if y = 1, I get x = 1
so we lucked out to find the intersection point as (1,1)

Wolfram confirmed this.

http://www.wolframalpha.com/input/?i=solve+y+%3D+x%5E3+%2B+14x+-+14+%3B+x+%3D+y%5E3+%2B+14y+-+14

it is not 14 it is 1/4

To find the number of points of intersection between a function and its inverse, we need to determine the points where the graph of the original function intersects the graph of its inverse. In this case, we need to find the points of intersection between the function f(x) = x^3 + (1/4)x - (1/4) and its inverse f^(-1)(x).

To begin, let's find the inverse of the function f(x). The inverse of a function is obtained by interchanging the x and y variables and solving for y.

Step 1: Replace f(x) with y.
y = x^3 + (1/4)x - (1/4)

Step 2: Interchange x and y.
x = y^3 + (1/4)y - (1/4)

Step 3: Solve the equation for y.
0 = y^3 + (1/4)y - (1/4) - x

Unfortunately, finding the inverse of this function analytically is quite complex. However, we can still determine the number of points of intersection between the function f(x) and its inverse by using graphical methods.

Plotting the graphs of f(x) and its inverse f^(-1)(x) on the same coordinate system, the number of points of intersection is equal to the number of points where the two graphs intersect.

By inspecting the graph, we can visually determine the number of intersections. Counting the points of intersection can be done by zooming in and examining the precision level of the graph.

If a more precise numerical answer is required, you can also use numerical methods such as Newton's method to find the intersection points between the two graphs.

So, the exact number of points of intersection between the function f(x) and its inverse f^(-1)(x) can be determined by graphically analyzing the plot or by using numerical methods if more precision is needed.

(Note: The equation mentioned does not follow a specific format. Please double-check the equation and context to ensure its correctness.)