Calculus question

Question

Sketch the graph of f(x)=x⁴-8x²+1. Identify symmetry, increasing and decreasing intervals, local extrema, concavity, inflection points, asymptotes, ans x and y interceptsif they exist.

Answer 1

f(x) = x⁴ - 8x² + 1

First, recognize the overall shape. The parent function is x⁴ , which is a symmetrical curve, opening up, with the minimum point at (0,1). At low values of x, the negative x2 term becomes dominant. This opens down.

To find the roots, we can "complete the square" for f(x):

f(x) = x⁴ - 8x² + 1 = 0

x⁴ - 8x² + 16 = 15

(x² - 4)² = 15

x² - 4 = ±√15

x = ±√(4 ± √15) = ±2.806, ±0.356 (ignoring the imaginary roots)

To find the minimums and maximums, calculate f'(x), and set = to 0

f'(x) = 4x³ - 16x = 0

x = 0, x = ±2

Finally, there is a concavity reversal between the minimums and the peak at x = 0

To find the concavity reversal, calculate the 2nd derivative, f"(x) and set = 0:

f"(x) = 12x² - 16 = 0

x² = 16/12 = 4/3

x = ±1.155

To tie all this together, use a graphing calculator or online graphing tool