Consider the function f(x)=12x^5+30x^4-160x^3+2

Question

The function has 3 critical numbers A<B<C:

A=

B=

C=

f(x) has inflection points at (reading from left to right) x= D, E, and F:

D=

E=

F=

Answer 1

Take the derivative to find the critical points:

f'(x) = 60x⁴ + 120x³ - 48x² and factor to get 12x²(5x² + 10x - 4)

Set it equal to zero to get critical points meaning x = 0 and (using the quadratic formula), x = -1 + 3√5/5 and x = -1 - 3√5/5. Butting these in order, left to right we get A = -1 - 3√5/5, B = 0, C = -1 + 3√5/5

To get the points of inflection, take the second derivative and set it equal to zero:

f""(x) = 240x3 + 360x2-96x factoring we get: 24x(10x2 + 15x -4) = 0 at x = 0 and (using the quadratic formula) x = (-15 + √385)/20 and x = (-15 - √385)/20. Putting those in order left to right we get D = (-15 - √385)/20, E = 0, F = (-15 + √385)/20