Question

consider continuous functions f, g, h, and k. then complete the statements.
function h is two times the square of the difference of x and 1.
k(x) = x⁴ + 2x² + 8x − 4
select the correct answer from each drop - down.
the function that has the least minimum value is function
the function that has the greatest minimum value is function

Explanation:

Step1: Find min of $f(x)$

From the graph, the vertex of $f(x)=x^2-2x-6$ is at $x=1$, $f(1)=-7$.

Step2: Find min of $g(x)$

From the table, the smallest value is $g(1)=-5$.

Step3: Find min of $h(x)$

$h(x)=2(x-1)^2$. The minimum is $0$ (when $x=1$).

Step4: Find min of $k(x)$

Take derivative: $k'(x)=4x^3+4x+8=4(x+1)(x^2-x+2)$. Set to 0: $x=-1$. $k(-1)=(-1)^4+2(-1)^2+8(-1)-4=1+2-8-4=-9$.

Step5: Compare all minima

Minima: $f=-7$, $g=-5$, $h=0$, $k=-9$.

Answer:

The function that has the least minimum value is function $k$.
The function that has the greatest minimum value is function $h$.