Question

How do you find the exact relative maximum and minimum of the polynomial function of f(x)=x³+4x²−5x?

Answer 1

To find the exact relative maximum and minimum of the polynomial function f(x) = x³ + 4x² − 5x, you need to take the derivative, evaluate the second derivative, and use the first derivative test.

Step-by-step explanation:

To find the exact relative maximum and minimum of the polynomial function f(x) = x³ + 4x² − 5x, we need to follow these steps:

1. Take the derivative of f(x) to find the critical points. The critical points are the points where the derivative is equal to zero or undefined.
2. Evaluate the second derivative of f(x) at the critical points to determine the concavity of the graph.
3. Use the first derivative test to find the relative extremum by analyzing the sign changes of the derivatives around the critical points.

For the given function, the first derivative is f'(x) = 3x² + 8x - 5 and the second derivative is f''(x) = 6x + 8. Using these derivatives, we can find the critical points, concavity, and relative extremum.