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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions5 Marks
Question
Determine the values of $x$ for which the function $f(x) = x^2 - 6x + 9$ is increasing or decreasing.
Also, find the coordinates of the point on the curve y = $x^2 - 6x + 9$ where the normal is parallel to the line $y = x + 5$.
✓

Answer

We have, $f(x) = x^2 - 6x + 9$
$\therefore$ $f'(x) = 2x - 6$
Critical points
$f'(x) = 0 \Rightarrow 2(x - 3) = 0 \Rightarrow x = 3$
Clearly, $f'(x) > 0 if x > 3 f'(x) < 0 if x < 3$
 Thus, f(x) is increases in $(3,\infty),$ decreases in $(-\infty,3)$
II part:
The given equation of curves $y = x^2 - 6x + 9 ....(i) y = x + 5 ....(ii)$
Slope of(i) $\text{m}_1=\frac{\text{dy}}{\text{dx}}=2\text{x}-6$
Slope of(ii) $\text{m}_2=1$
Given that slope of normal to (i) is parallel to (ii) $\therefore\ \frac{-1}{2\text{x}-6}=1$
$\Rightarrow2\text{x}-6=-1$ $\Rightarrow\text{x}=\frac{5}{2}$ From (i) $\text{y}=\frac{25}{4}-15+9$
$=\frac{25}{4}-6$ $=\frac{1}{4}$ Thus, the required point is $\Big(\frac{5}{2},\frac{1}{4}\Big).$

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