Untersuchung der Funktion $h(x) = 0,5e^{0,5x} - x + 1,5$

In this problem, we are given a function h of x and need to analyze its graph, find its extremum, perform shifts, and check a tangent property. Let's start with the definition of the function.

For part three point two, we need to calculate the coordinates of the extreme point of h. This requires finding the derivative and setting it to zero.

The derivative h prime of x is found using the chain rule for the exponential part. The derivative of zero point five e to the power of zero point five x becomes zero point two five e to the power of zero point five x. The derivative of negative x is negative one.

We set the derivative to zero to find the stationary point. Adding one and multiplying by four, we get e to the power of zero point five x equals four.

Taking the natural logarithm of both sides, zero point five x equals the natural log of four. Thus, x equals two times the natural log of four, which simplifies to approximately two point seven seven.

Now, let's find the y-coordinate by plugging this back into h. Since e to the power of zero point five x is four, the first term becomes zero point five times four, which is two.

By checking the second derivative, we confirm this is a minimum. The coordinates are approximately two point seven seven and zero point seven three.

Now for part a, we shift the graph in the y-direction so it passes through the origin. This means h of zero plus the shift must be zero.