Analysis und Transformation einer Exponentialfunktion

Today we will analyze the function h of x given this exponential term minus x plus one point five. We will find its extreme point, shift its graph, and check the property of a tangent line.

In part three point two, we need to find the coordinates of the extreme point. To do this, we first find the derivative of h.

The derivative h prime of x is zero point five times zero point five times e to the power of zero point five x, minus one.

Simplifying the coefficient, we get zero point two five times e to the zero point five x minus one.

To find the extreme point, we set the derivative equal to zero.

Add one to both sides, then divide by zero point two five, which is the same as multiplying by four.

Now we take the natural logarithm of both sides to solve for x.

Multiplying by two gives us x equals two times the natural logarithm of four. Using the properties of logs, this is log of sixteen, which is about two point seven seven.

Next, we calculate the y-coordinate by plugging this x back into the original function h.

This simplifies to zero point five times four minus two log four plus one point five, which results in three point five minus log sixteen, roughly zero point seven three.