Question

the polynomial function $g$ is defined by $g(x) = -4x^4 - 9x^3 + 2x^2 + 7x - 2$.
use the aleks graphing calculator to find all the points $(x, g(x))$ where there is a local minimum.
round to the nearest hundredth.
if there is more than one point, enter them using the \and\ button.
$(x, g(x)) = (-0.67, -0.73)$
and $(x, g(x)) = (1.00, -6.00)$

Explanation:

Step1: Find derivative of $g(x)$

$g'(x) = -16x^3 -27x^2 +4x +7$

Step2: Solve $g'(x)=0$ for critical points

Using a graphing calculator, the critical $x$-values are approximately $x\approx-1.89$, $x\approx-0.67$, $x\approx1.00$

Step3: Test intervals for local minima

Using the first derivative test:

• For $x\approx-0.67$: The derivative changes from negative to positive, so this is a local minimum. Calculate $g(-0.67)\approx-0.73$.
• For $x\approx1.00$: The derivative changes from negative to positive, so this is a local minimum. Calculate $g(1.00)=-4(1)^4 -9(1)^3 +2(1)^2 +7(1)-2=-4-9+2+7-2=-6.00$.
• For $x\approx-1.89$: The derivative changes from positive to negative, so this is a local maximum, not a minimum.

Answer:

$(x, g(x)) = (-0.67, -0.73)$ and $(x, g(x)) = (1.00, -6.00)$