<s>[INST] minimum value of $f(t) = 10t^6-24t^5+15t^4+40t^2+108$ without derivative minimum value of $f(t) = 10t^6-24t^5+15t^4+40t^2+108$ without derivative
for $t\leq 0$ value of function $f(t)\geq 108$
i wan,t be able to proceed after that ,could some help me with this
 [/INST] Note that 
\begin{align}
f(t) &= 10t^6-24t^5+15t^4+40t^2+108
\\
&= 10t^4\left(t^2-\dfrac{12}{5}t+\dfrac{3}{2}\right)+40t^2+108
\\
&= 10t^4\left(t^2-\dfrac{12}{5}t+\dfrac{36}{25}+\dfrac{3}{50}\right)+40t^2+108
\\
&= 10t^4\left(\left(t-\dfrac{6}{5}\right)^2+\dfrac{3}{50}\right)+40t^2+108.
\end{align}
Now, can you show that $f(t) \ge 108$ for all real $t$?
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