Consider that an idela gas (n mole) is espanding in a process gives by `p=f(v)`, which passes throught a point `(V_(0),p_(0))`. Shown that the gas is absorbing heat at `(p_(0),v_(0))` if the slope of the curve `p=f(v)` is large than the slope of the adabatic passing throught `(p_(0), v_(0))`.

Explanation

To solve the problem, we need to show that an ideal gas is absorbing heat at the point \((P_0, V_0)\) if the slope of the curve \(p = f(v)\) is greater than the slope of the adiabatic process passing through the same point. We will use the first law of thermodynamics and the definitions of the slopes involved.

### Step-by-Step Solution:

1. **Identify the Slope of the Given Process**:
The process is defined by the equation \(p = f(v)\). At the point \((P_0, V_0)\), the slope of this curve is given by:
\[
\frac{dp}{dv} = f'(V_0)
...