Given The Function 8x 2-7x 3 Calculate The Following Values

The function 8x² - 7x³ is a cubic polynomial that combines both quadratic and cubic terms. Calculating specific values of this function is essential for understanding its behavior, finding roots, and applying it to real-world problems.

Understanding the Function

The function f(x) = 8x² - 7x³ can be analyzed by examining its components:

f(x) = -7x³ + 8x²

This is a cubic polynomial with a leading coefficient of -7 and a quadratic term.

Key Characteristics

Degree 3 polynomial (cubic)
Leading term determines end behavior
Can have up to 3 real roots
Concave down for x > 4/7

End Behavior

As x approaches positive infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches positive infinity.

Calculating Specific Values

To calculate specific values of the function, simply substitute the x-value into the equation:

f(x) = -7x³ + 8x²

Example Calculation

Let's calculate f(2):

f(2) = -7(2)³ + 8(2)² = -7(8) + 8(4) = -56 + 32 = -24

So, when x = 2, the function value is -24.

Common Values Table

x	f(x)
0	0
1	1
2	-24
3	-27
-1	-16

Graphing the Function

Graphing the function helps visualize its behavior. Key points to plot include:

Roots (where f(x) = 0)
Critical points (where f'(x) = 0)
Inflection points
Interesting x-values like -1, 0, 1, 2, 3

The graph will show the cubic nature of the function with a characteristic "S" shape.

Practical Applications

This function appears in various mathematical and real-world contexts:

Optimization problems
Physics equations
Economic modeling
Engineering calculations

Understanding how to calculate and interpret values of this function is valuable across many disciplines.

Frequently Asked Questions

How do I find the roots of the function?

To find the roots, solve the equation -7x³ + 8x² = 0. Factor out x² to get x²(-7x + 8) = 0. This gives roots at x = 0 and x = 8/7.

What is the derivative of this function?

The derivative is f'(x) = -21x² + 16x. This helps find critical points and analyze the function's increasing/decreasing behavior.

How does the function behave as x approaches infinity?

As x approaches positive infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches positive infinity.