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

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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.
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Hard to tell if you did it right without knowing what you did.
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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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