Solve each problem. Give the maximum number of turning points of the graph of each function. ƒ(x)=4x^3-6x^2+2

Verified step by step guidance

1. Identify the degree of the polynomial. The function \( f(x) = 4x^3 - 6x^2 + 2 \) is a cubic polynomial, which means it has a degree of 3.
2. Recall that the maximum number of turning points of a polynomial function is one less than its degree. Therefore, for a cubic polynomial, the maximum number of turning points is \(3 - 1 = 2\).
3. Understand that turning points are where the graph changes direction from increasing to decreasing or vice versa.
4. Note that the actual number of turning points can be less than the maximum, depending on the specific function.
5. Conclude that the maximum number of turning points for the function \( f(x) = 4x^3 - 6x^2 + 2 \) is 2.

Key Concepts

Turning Points

Turning points of a function are points where the graph changes direction from increasing to decreasing or vice versa. These points are critical for understanding the shape of the graph and can be found by analyzing the first derivative of the function. A function can have a maximum of n-1 turning points, where n is the degree of the polynomial.

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The degree of a polynomial, which is the highest power of the variable, determines its general shape and the maximum number of turning points. In this case, the function ƒ(x)=4x^3-6x^2+2 is a cubic polynomial.

First Derivative Test

The first derivative test is a method used to find the critical points of a function, which are points where the derivative is zero or undefined. By setting the first derivative equal to zero, we can identify potential turning points. Analyzing the sign of the derivative around these points helps determine whether they are local maxima, minima, or neither.