show constant k is mean value of f between closed interval [a,b]

For each of the two integrals: integrate each function separately. Meaning, you can write:

int{a,b}(f(x)-k) = int{a,b}(f(x)) - int{a,b}(k)

Now you just need to integrate k (a constant function), and then you can solve for k.
Hard to tell if you did it right without knowing what you did.
So I$f(x)$-I$k$=I$k$-If(x)\]

2I$f(x)$=2I$k$

I$f(x)$=I$k$

If the area under the curve equals the area of the rectangle base $a,b$

can you convince yourself the height of the rectangle (k) must equal the mean value of f?

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