Question

Use a computer algebra system to graph f and to find f ' and f ''. Use graphs of these derivatives to find the following. (Enter your answers using interval notation. Round your answers to two decimal places.)

f(x) =

x3 + 5x2 + 1

x4 + x3 − x2 + 2

The intervals where the function is increasing.

The intervals where the function is decreasing.

The local maximum values of the function. (Enter your answers as a comma-separated list.)

The local minimum values of the function. (Enter your answers as a comma-separated list.)

The inflection points of the function.

The intervals where the function is concave up.

The intervals where the function is concave down.

Answer 1

Increasing Intervals: The derivative of a function may be used to determine whether the function is increasing at any intervals in its domain. If f′(x) > 0, then f is increasing on the interval.

Decreasing Intervals: The derivative of a function may be used to determine whether the function is decreasing at any intervals in its domain. If f′(x) < 0, then f is decreasing on the interval.

Maximum Values: The largest value of the function f(x) on the entire domain of a function.

Minimum Values: The smallest value of the function f(x) on the entire domain of a function.

Inflections Points: An inflection point is a point on the graph of a function at which the concavity changes. Points of inflection can occur where the second derivative is zero. i.e.f''(x)=0.

Concave Upward: When the function y = f (x) is concave up, the graph of its derivative y = f '(x) is increasing. If f''(x)<>0 then f(x) concave downwards.

Concave Downward: When the function y = f (x) is concave down, the graph of its derivative y = f '(x) is decreasing. If f''(x)< 0 then f(x) concave downwards.