The polynomial $G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x$ measures the concentration of a dye in the bloodstream $x$ seconds after it is injected. Does the concentration increase between 14 and 15 seconds?

Question

Quizplus · Accepted Answer

The concentration is increasing if the derivative is positive. The derivative is $G'(x) = -0.024x^3 + 0.42x^2 - 1.06x + 1.79$. At $x=14$, $G'(14) = -0.024(14)^3 + 0.42(14)^2 - 1.06(14) + 1.79 = 0.014 > 0$. So the concentration is increasing at $x=14$. At $x=15$, $G'(15) = -0.024(15)^3 + 0.42(15)^2 - 1.06(15) + 1.79 = 0.029 > 0$. So the concentration is increasing at $x=15$. Since the concentration is increasing at both $x=14$ and $x=15$, the concentration increases between 14 and 15 seconds.

The Polynomial G(x)=−0.006x4+0.140x3−0.53x2+1.79xG ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 xG(x)=−0.006x4+0.140x3−0.53x2+1.79x

The Polynomial $G ( x ) = - 0.006 x ^ { 4 } + 0.140 x ^ { 3 } - 0.53 x ^ { 2 } + 1.79 x$