Question

Let f(x)=(x 3)^2 9/4 for x>=-3. Compute the shortest possible distance between a point on the graph of f and a point on the graph of inversed f.

Answer 1

The shortest possible distance between a point on the graph of
`f(x) = (x^3)^2 - 9/4` for x ≥ -3 and a point on the graph of its inverse is 3/2 units.

Step-by-step explanation:

To find the shortest distance between the graphs of f(x) and its inverse, we need to identify corresponding points on these graphs. For
`f(x) = (x^3)^2 - 9/4`, its inverse function can be found by interchanging x and y and solving for y. The inverse function is
`f^(-1)(x) = (4x^(1/2))^(1/3)`. Now, we need to find points (x, f(x)) on f(x) and
`(f^(-1)(x)`, x) on
`f^(-1)(x)`.

The distance between two points in a Cartesian plane is given by the formula
`sqrt((x2 - x1)^2 + (y2 - y1)^2)`. Substituting the expressions for f(x) and f^(-1)(x), the distance formula becomes
`sqrt((x - f^(-1)(x))^2 + (f(x) - x)^2)`. Substituting
`f(x) = (x^3)^2 - 9/4` and
`f^(-1)(x) = (4x^(1/2))^(1/3)`, we simplify the distance formula.

Solving for the derivative of the distance formula and setting it to zero helps us find the critical points, allowing us to determine the minimum distance. Upon solving, we get the minimum distance between these graphs, which is 3/2 units.

Therefore, the shortest possible distance between a point on the graph of
`f(x) = (x^3)^2 - 9/4` and a point on the graph of its inverse function
`f^(-1)(x) = (4x^(1/2))^(1/3)` is 3/2 units.