Steckbriefaufgaben und bestimmte Integrale

In this exercise, we will solve two problems. First, we will set up a system of linear equations for a polynomial of degree four, and then we will solve a definite integral for an unknown upper limit.

Let's start with task one point five. We are looking for the general form of a polynomial function of degree four.

Since many conditions involve the slope, we also need the first derivative.

We have five unknowns, a through e, so we need five pieces of information. Let's list the given points and properties.

Now, let's substitute these conditions into our general formulas to build the system of equations.

Condition one: f of zero equals four. Plugging zero into the formula, all terms with x vanish, leaving e equals four.

Condition two: f prime of zero equals zero because we have a local maximum there. This tells us d must be zero.

Next, the point T lies on the graph, so f of one equals two. Substituting one for x gives us this equation.

Since T is a local minimum, the derivative at x equals one is also zero.

Finally, the slope at negative one is twelve. Plugging negative one into the derivative formula, we get negative four a plus three b minus two c plus d equals twelve.