Question

Let f be the function given by f(x)=1/7 x^7+1/2 x^6-x^5-15/4 x^4+4/3 x^3+6x^2. Which of the following statements is true?

A) f'(-3.1) < f'(-1.5) < f'(0.4)
B) f′(−3.1) < f′(0.4) < f′(−1.5)
C) f′(−1.5) < f′(0.4) < f′(−3.1)
D) f′(0.4) < f′(−1.5) < f′(−3.1)

Answer 1

f(x) = 1/7 x^7 + 1/2 x^6 -x^5 - 15/4 x^4 + 4/3 x^3 + 6x^2

f'(x) = x^6 + 3x^5 - 5x^4 - 15x^3 + 4x^2 + 12x
= (x+3)(x+2)(x+1)x(x-1)(x-2)
so
f'(x) >0 on (-∞,3)U(-2,-1)U(0,1)U(2,∞)
f'(x) < 0 on (-3,-2)U(-1,0)U(1,2)
so now you can answer the question.