greatest value of function depends on parameter $k$

Question

Find the greatest value of the function

$f(x)=x^4-6kx^2+k^2$on the interval $[-2,1]$

depending on the parameter $k$

My Try: $$f(x)=x^4-6kx^2+9k^2-8k^2$$

$$f(x)=(x^2-k)^2-8k^2$$

from $-2 \leq x\leq 1\Rightarrow 0 \leq x^2\leq 4$

$$-7k^2\leq (x^2-k)^2-8k^2 \leq (4-k)^2-8k^2$$

Could some help me to solve it , Thanks

Answer 1

Let $x^2=t$ and $g(t)=t^2-6kt+k^2.$

Thus, $0\leq t\leq4$ and since a graph of $g$ it's a parabola as $\cup$, we obtain: $$\max_{[-2,1]}f=\max_{[0,4]}g=\max\{g(0),g(4)\}=\max\{k^2,k^2-24k+16\}.$$