Polinom Grafikleri ve Katsayı Bulma

Let's analyze this polynomial geometry problem. We are given the function P of x equals x to the fourth minus four x cubed plus a x squared.

First, notice that P of x passes through the origin because every term has an x. Also, let's find the critical points by taking the derivative.

We can factor out a 2x. This reveals that x equals zero is always a critical point, corresponding to the local maximum we see in the red graph's center.

The other critical points are the roots of the quadratic part. From Vieta's formulas, the sum of these roots is negative b over a, which is 6 divided by 2, equalling 3.

Looking at the handwritten notes on the graph, we see markings 'minus 1' and '2'. Let's check if the roots could be -1 and 4, which sum to 3.

If the roots are -1 and 4, then the product of roots, a over 2, must equal negative 4. This implies a equals negative 8.

Let's assume a is negative 8 and verify the shape. The polynomial becomes x fourth minus 4 x cubed minus 8 x squared.

This function has a W-shape. The local maximum is at x equals 0 with value 0. The global minimum will be at the critical point further from the origin, x equals 4.