Find the minimum value of a, function f(x)=x^2 +a x+10 is increas… - Vidyadip

Question

Find the minimum value of a, function $f(x)=x^2$ $+a x+10$ is increasing in $[3,6]$.

✓

Answer

Given,
$
\begin{array}{rr}
& f(x)=x^2+a x+10 \\
\text { So, } & f^{\prime}(x)=2 x+a \\
\text { because } & x \in[3,6] \Rightarrow 3 \leq x \leq 6 \\
\Rightarrow & 6 \leq 2 x \leq 12 \\
\Rightarrow & 6+a \leq 2 x+a \leq 12+a \\
\Rightarrow & 6+a \leq f^{\prime}(x) \leq 12+a
\end{array}
$
If $f(x)$ is increasing then $f^{\prime}(x)>0$$
\Rightarrow \quad 6+a>0 \Rightarrow a>-6
$
So, minimum value of a is -6 .

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