Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=2x⁴

Verified step by step guidance

1. Identify the function given: \(f(x) = 2x^4\). This is a polynomial function, and its domain is all real numbers, \((-\infty, \infty)\).
2. Find the first derivative of the function to determine where it is increasing or decreasing. Use the power rule: \(f'(x) = \frac{d}{dx}(2x^4) = 8x^3\).
3. Set the derivative equal to zero to find critical points: \$8x^3 = 0\(. Solve for \)x$ to find critical points where the function could change from increasing to decreasing or vice versa.
4. Determine the sign of \(f'(x)\) on intervals defined by the critical points. Test values in each interval to see if \(f'(x)\) is positive (function increasing) or negative (function decreasing).
5. Based on the sign of \(f'(x)\), write the largest open intervals where \(f(x)\) is increasing or decreasing. Remember, if \(f'(x) > 0\) on an interval, \(f(x)\) is increasing there; if \(f'(x) < 0\), \(f(x)\) is decreasing.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like ƒ(x) = 2x^4, the domain is all real numbers since polynomials are defined everywhere on the real line.

Increasing and Decreasing Intervals

A function is increasing on an interval if its output values rise as the input values increase, and decreasing if the output values fall. Identifying these intervals involves analyzing the behavior of the function’s graph or its derivative to see where the slope is positive or negative.

Using the Derivative to Determine Monotonicity

The derivative of a function gives the slope of the tangent line at any point. By finding where the derivative is positive, negative, or zero, we can determine intervals where the function is increasing, decreasing, or has critical points, respectively. For ƒ(x) = 2x^4, the derivative helps identify these intervals.