Calculus Question

Question

Determine where the curve y = x³+4x²-2x+1 is concave up and concave down.

 

Help!?

Answer 1

Set the second derivative of f(x) equal to zero.

 

f'(x) = 3x² + 8x - 2

 

f"(x) = 6x + 8

 

6x + 8 = 0

 

2(3x + 4) = 0

 

x = -4/3 = -1.33

 

The second derivative is negative in the interval (-∞, -4/3).  Concave down in this interval.

The second derivative is positive in the interval (-4/3, ∞).  Concave up in this interval.

Answer 2

Find the second derivative of y.  Where it is positive, the curve is concave up.  Where it is negative, the curve is concave down.

 

y = x³ + 4x² - 2x + 1

y' = 3x² + 8x - 2

y'' = 6x + 8

 

y'' = 0 at x = -8/6 = -4/3

 

If x < -4/3, y'' < 0 and the curve is concave down.  If x > -4/3, y'' > 0 and the curve is concave up.