The graph of \(f\) on the interval \([0,5]\) is given below.

Using geometry, we can evaluate \(\int _0^5 f(x) \d x\) to find it equals \(\answer {6.5}\).

Again, using geometry, we can evaluate \(\int _1^4 f(x) \d x\) to find it equals \(\answer {4.5}\).

Finally, we can use geometry to find that \(\int _0^2 4f(x) \d x - \int _1^3 5f(x) \d x = \answer {-6.5}\).

The average value of \(f\) on \([0,5]\) is

\[ \bar {f} = \answer {1.3}. \]

The points \(c\) in \([0,5]\) where \(f(c) = \bar {f}\) are

\[ c_1=\answer {2.3} \]
and
\[ c_2=\answer {4.7}, \]
(answer from left to right).

Since \(1<\bar {f}<2\), the point \(c\) will be found only at the intervals where \(1<f(x)<2\).

By inspecting the graph of \(f\), we can see that the only such intervals are \((2,3)\) and \((4,5)\).

From the graph we can conclude that

\(f(x)=x-1\), for \(2<x<3\); and that

\(f(x)=6-x\), for \(4<x<5\).

Now we can solve the equation

\[ \bar {f} =f(c). \]