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  4. What is the least value of k such that the function x2 + kx + 1 is strictly increasing on (1,2)

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Q. What is the least value of k such that the function x$^2$ + kx + 1 is strictly increasing on (1,2)

VITEEEVITEEE 2006

Solution:

Let f(x) = x$^2$ + kx + 1
f '(x) = 2x + k
f(x) is strictly increasing on (1, 2)
if f '(x) > 0 for x $\in $(1, 2)
$\Rightarrow $ 2x + k > 0 for x $\in $(1, 2)
$\Rightarrow $ k > -2x for x $\in$ (1, 2)
Now, 1 < x < 2 $\Rightarrow $ 2 < 2x < 4
$\Rightarrow $ -2 > -2x > -4
$\Rightarrow $ - 4 < -2x < -2
$\Rightarrow \, \, k \ge \, \, -2$
Hence least value of k = -2.