What are the true characteristics of the graph of the function f(x) = 6x^4 – x^2?

The function f(x) = 6x^4 – x^2 is a polynomial function characterized by a few key traits:

• Degree and Leading Coefficient: The degree of the function is 4 (the highest exponent of x), and the leading coefficient is 6, which is positive. This indicates that as x approaches positive or negative infinity, the graph of the function will rise towards infinity, meaning both tails of the graph will extend upwards.
• End Behavior: Given that it’s a fourth-degree polynomial with a positive leading coefficient, we know that the graph will approach infinity (∞) as x approaches both positive and negative infinity.
• Critical Points: To determine where the function has critical points (where the graph could have maxima or minima), we find the derivative of the function:
  f'(x) = 24x^3 - 2x. Setting the derivative equal to zero gives us 24x^3 - 2x = 0, which simplifies to 2x(12x^2 - 1) = 0. This means that x = 0 or x = ±1/√12 (or approximately ±0.2887).
• Local Maxima and Minima: Evaluating the second derivative, f''(x) = 72x^2 - 2, at the critical points will help determine the concavity of the graph and whether those points are local maxima or minima.
• X-Intercepts: To find the x-intercepts, we set the function equal to zero: 6x^4 - x^2 = 0. Factoring it out gives x^2(6x^2 - 1) = 0. Thus, the solutions are x = 0 and x = ±√(1/6).
• Y-Intercept: The y-intercept occurs when x = 0. Plugging this into the function gives f(0) = 0, so the graph crosses the y-axis at (0,0).

In summary, the graph of the function f(x) = 6x^4 – x^2 is a quartic polynomial that rises on both ends, has a y-intercept at (0,0), and contains both local maxima and minima at determined critical points.