Q. 25.0( 1 Vote )

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 :

Given:- Function f(x) = x2 – 6x + 9 and a line parallel to y = x + 5


Theorem:- Let f be a differentiable real function defined on an open interval (a, b).


(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)


(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)


Algorithm:-


(i) Obtain the function and put it equal to f(x)


(ii) Find f’(x)


(iii) Put f’(x) > 0 and solve this inequation.


For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.


Here we have,


f(x) = x2 – 6x + 9



f’(x) = 2x – 6


f’(x) = 2(x – 3)


For f(x) lets find critical point, we must have


f’(x) = 0


2(x – 3) = 0


(x – 3) = 0


x = 3





clearly, f’(x) > 0 if x > 3


and f’(x) < 0 if x < 3


Thus, f(x) increases on (3, ∞)


and f(x) is decreasing on interval x (–∞, 3)


Now, lets find coordinates of point


Equation of curve is


f(x) = x2 – 6x + 9


slope of this curve is given by




m1 = 2x – 6


and Equation of line is


y = x + 5


slope of this curve is given by




m2 = 1


Since slope of curve (i.e slope of its normal) is parallel to line


Therefore, they follow the relation




2x – 6 = –1



Thus putting the value of x in equation of curve, we get


y = x2 – 6x + 9






Thus the required coordinates is )



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