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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions4 Marks
Question
Determine the values of x for which the function f(x) = x2 - 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 - 6x + 9 where the normal is parallel to the line y = x + 5.
✓

Answer

We have,

f(x) = x2 - 6x + 9

$\therefore$ f'(x) = 2x - 6

Critical points

f'(x) = 0

⇒ 2(x - 3) = 0

⇒ 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 = x2 - 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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