Find the open interval on which the function f(x)=x^4-3x^2+5x is increasing or decreasing

Question

Find the open interval on which the function f(x)=x^4-3x^2+5x is increasing or decreasing

Answer 1

A function is increasing when the derivative is positive and decreasing when the derivative is negative.

To find the derivative of f(x) = x⁴ - 3x² + 5x, use the power rule:

f '(x) = 4x³ - 6x + 5

To find the boundary between positive and negative, set this equal to zero and solve. The algebraic solution to 4x³ - 6x + 5 = 0 is very difficult to do, so the best way to solve it is to use your graphing calculator. there is only 1 solution, x = -1.523340162075

You can now put this on a number line and evaluate the derivative for a point less than this and for a point greater than is.

A random point less is -2:

f '(-2) = -15 therefore the function is decreasing on (-∞, -1.523340162075)

A random point greater is -1:

f '(-1) = 7 therefore the function is increasing on (-1.523340162075, ∞)

Answer 2

To determine the regions in which f(x) is increasing or decreasing, we take its derivative to find an equation for the slope of f(x). Use the power rule (multiply the leading coefficient of each term by its exponent, then reduce the exponent by one)

f'(x) = d/dx (x⁴ - 3x² + 5x) = 4x³ - (2*3) x¹ + 5x⁰ = 4x³ - 6x + 5

Now that we have the derivative, we can factor it. Factoring a polynomial gives us the points at which the equation is equal to zero. For a derivative of a function f, the zeroes (x-values) symbolize the points where f is "flat". In other words, the zeroes of a derivative indicate the boundaries between the original function increasing or decreasing.

Now, you could try to use the [Rational Root Theorem](https://en.wikipedia.org/wiki/Rational_root_theorem) to get an exact answer, but this polynomial only has an irrational root so you would not get any where. One thing you can try is the [Cubic Formula](https://en.wikipedia.org/wiki/Cubic_equation#Cardano's_formula), which involves a lot of ugly algebra but will give you the exact irrational solution.

If you had this question on an exam you would probably have a calculator with you to avoid having to do that. The zero is approximately -1.52334.

Now since we have only one zero of the derivative, that means the function must be increasing on one side and decreasing on the other. We just have to the derivative f'(x) with input x-values on either side of the above value. For example, let's try x = -2 and x = 0:

f'(-2) = 4 * (-8) - 6 * (-2) + 5 = -32 + 12 + 5 = -15

f'(0) = 4 * 0 - 6 * 0 + 5 = 5

Since the derivative changes sign from negative to positive, f(x) must have positive slope (and is increasing) after x = -1.52334. If x is less than -1.52334, then f(x) is decreasing.