# Mathematics - Calculus - Functions

Given the function f(x) = x^3 -3x-1 solve the following algebraically must show work algebraically

Identify the intervals on which the following are true

(1) Function is increasing

(2) Slopes of tangent lines are positive

(3) Function is concave up

(4) Slopes of tangent lines are increasing ie becoming less negative or more positive

(5) Function is decreasing

(6) Slopes of tangent lines are negative

(7) The function is concave down

(8) The slopes of the tangent lines are decreasing ie becoming less positive or more negative

(9) The rate of change is decreasing

(10) Find the equation of the line tangent to the graph at its point of inflection

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#### Solution Preview

Please see the attachment.

f(x) = x^3 - 3x - 1

f'(x) = 3x^2 - 3 = 3(x + 1)(x - 1) and f "(x) = 6x

(1) f"(x) > 0 ïƒ 3(x + 1)(x - 1) > 0 ïƒ x E (-infinity, -1) U (1, infinity).

(2) f'(x) itself is the slope of the tangent ...

#### Solution Summary

This solution contains step-by-step calculations to solve the given function algebraically and it also identifies if a statement are true or false.