Consider the followingg f(x)=4x^3+9x^2-54x+6

Question

a)find interval of increase/ decrease

b)find local min/max

c) Find inflection point()

and interval notation where its concave up/down

I got it started with getting derivative first

f''(x)=12x^2+18x-54 but then when it turned to 6(2x^2+3x-9) does that parenthesis part then get set equal to zero after?

Answer 1

a) f'(x) = 12x² + 18x - 54 = 0; 2x²+ 3x - 9 = 0; (2x - 3)(x + 3) = 0, x = - 3, x = 1.5

f'(x) > 0 (- ∞, -3), (1.5, ∞) f(x) increasing

f'(x) < 0, (- 3, 1.5) f(x) decreasing

b) (-3, 951) is local max because f'(x) changes sign from + to -

(1.5, 161) is local minimum because f'(x) change sign from - to +

c) f''(x) = 24x + 18

f''(x) > 0, 24x + 18 > 0; x > - 3/4, (- 3/4, ∞), f(x) concave up;

f''(x) = 24x + 18 < 0; x < - 3/4; (- ∞, -3/4) f(x) concave down

(- 3/4, 100.5) is inflection point