Calculus Question

Question

Let f(x)=1x3−2x2−5x+6. Find the interval where f is concave up. If an end-point of the interval is either −∞ or +∞ type "infinity" (without quotation marks).  Answer: _< X <__

Find the interval where f is concave down. If necessary, indicate −∞ or +∞ by typing "infinity" (no quotation marks). Answer: _ < X <__

Find the inflection point of f. Answer: x=

Answer 1

note that x^3 - 2x^2 -5x + 6 = (x-3)(x - 1)(x + 2) Roots are 3, 1, and -2. We note that the function will start off concave down and then turn convex (or concave up). If the roots were 3, 1, and -1, the switch from concave to convex would be at 1, which tells us that the solution will be a little less than 1.

The first derivative is 3x^2 - 4x - 5

The second derivative is 6x - 4

6x - 4 < 0 if x < 2/3

Concave on (-infinity, 2/3). Note how 2/3 is a little less than 1.

Answer 2

f(x) = x³-2x²-5x+6

Find the critical points

f'(x) = 3x²-4x-5

Setting f'(x) to zero and solving for x

x=2.12, -0.78

f(2.12) = -4.06 local min

f(-0.78) = 8.21 local max

f(x) is concave down -infinity < x <2.12

f(x) is concave up -0.78 < x < infinity