Download App

APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 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 $\Big(\frac{5}{2},\frac{1}{4}\Big).$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integral:
$\int\frac{\text{x}^4+1}{\text{x}^2+1}\text{ dx}$
→
Find the area bounded by the curve $\text{y}=\cos\text{x}$, x-axis and the ordinates $\text{x}=0, \text{x}=2\pi$.
→
A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission.
Month
Sale of units
Total commission drawn (in Rs.)
 
A
B
C
 
Jan
90
100
20
800
Feb
130
50
40
900
March
60
100
30
850
 
Find out the rates of commission on items A, B and C by using determinant method.
→
If $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=0,$ then show that $\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{b}}\times\vec{\text{c}}=\vec{\text{c}}\times\vec{\text{a}}.$ Interpret the result geometrically?
→
Draw a rough sketch of the region bounded by the parabola $y^2 = 4x$ and $x^2 = 4y$ by using methods of integration.
→
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12cm is 16cm.
→
Two schools $P$ and $Q$ want to award their selected students on the values of Tolerance, Kindness, and Leadership. The school $P$ wants to award $Rs. x$ each, $Rs. y$ each and $Rs. z$ each for the three respective values to $3, 2$ and $1$ students respectively with total award money of $Rs. 2200$.
School $Q$ wants to spend $Rs. 3100$ to award its $4, 1$ and $3$ students on the respective values $($by giving the sameaward money to the three values as school $P)$. If the total amount of award for one prize on each value is $Rs.1200,$ using matrices, find the award money for each value
→
Evaluate the following intregals:
$\int\frac{\text{x}+2}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}$
→
Differentiate the following functions with respect to x:
$\sin^2\{\log(2\text{x}+3)\}$
→
Find the differential equation of the family of curves $\text{(x- h)}^{2} + \text{(y - k)}^{2} = \text{r}^{2}, $ where h and k are arbitrary constants.
→