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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions3 Marks
Question
State when a function $f(x)$ is said to be increasing on an interval $[a, b].$ Test whether the function $f(x) = x^2 - 6x + 3$ is increasing on the interval $[4, 6].$
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Answer

A function $f(x)$ is said to be increasing on an interval $[a, b]$ if it is increasing at $x = a$ and $x = b.$
Here,
$f(x) = x^2 - 6x + 3$
$f'(x) = 2x - 6$
$f'(x) = 2(x - 3)$
Now,$ f'(4) = 2(4 - 3)$
$= 2$
$\therefore f'(4) > 0$
So, $f(x)$ is increasing on $x = 4$
$f'(6) = 2(6 - 3)$
$= 6$
$\therefore f'(6) > 0$
So, $f(x)$ is increasing on $x = 6$
Hence, $f(x)$ is increasing on $[4, 6].$

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