For what values of a the function f given by f(x) = x^2 + ax + 1 … - Vidyadip

Question

For what values of a the function f given by $f(x) = x^2 + ax + 1$ is increasing on $[1, 2]$?

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Answer

It is given that function $f(x) = x^2 + ax + 1$
$f^{\prime}(x)=2 x+a$
Now, function $f$ will be increasing in $[1,2]$,
$\text { if } f^{\prime}(x)>0 \text { in }[1,2]$
$\Rightarrow 2 x+a>0$
$\Rightarrow 2 x>-a$
$\Rightarrow a<-2 x$
Therefore, we have to find the least value of a such that
$\Rightarrow a<-2 x \text { when } x \in[1,2]$
Now, $1 \leq \mathrm{x} \leq 2$
$\Rightarrow-4 \leq-2 x \leq-2$
Therefore, the least value of a for $f$ to be increasing on $[1,2]$ is given by
$\Rightarrow a=-4$
Therefore, the least value of $a$ is -4

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