Let $X$ be a single observation from the density $f(x) = (2θx + 1 − θ)I[0,1](x)$ with $−1≤ θ ≤ 1$. Find the most powerful test of size $α$ and its power
 Find a most powerful test of size $?$ of $H_0 : ? = 0$ versus $H_1 : ? = 1$.
 Find a uniformly most powerful test of size $?$ of $H_0 : ? = 0$ versus $H_1 : ? > 0$
 To test $H_0: ? ? 0$ versus $H_1: ? > 0$, the following procedure was used: Reject $H_0$ if $X$ exceeds $1/2$. Find the power and size of this test.

I feel the bounty is a little to low for this.

I second that.

Bounty has been increased

Is this for only one observation (n=1) or for a general random sample of n observations?

@Rage, this is a good question, yes the original question states let X be a single observation from the density. I will update the question to make this clear.

Without that hypothesis the problem was too difficult!
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Hi, just checking on page 2, the derivative (h'(x)) should the second term be 2θb(2θax+12θa) instead of 2θ b(2θax+12θb)?

Yes, there was a typo in the solution. The final result does not change, though. Let me add a new version.
The answer is accepted.
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