The following results are very useful in solving rational algebraic operations:
a) ab>0=> (a>0 and b>0) or (a<0 & b<0)
b) ab<0=> (a>0 and b<0) or (a<0 & b>0)
c) ab>0 and a>0 => b>0
d) ab<0 and a<0 => b>0
If P(x) and Q(x) are polynomial, then the Inequations P(x)/Q(x) >0, P(x)/Q(x) <0, P(x)/Q(x) ≥ 0 and P(x)/Q(x) ≤0 are known as rational algebraic Inequations. These Inequations can be solved by using the following STEPS
Step 1: Factorize P(x) and Q(x) into linear factors.
Step 2: Make Coefficient of x positive in all factors.
Step 3: Equate all the factors to zero and find the corresponding values of x. These values are generally known as critical points.
Step 4: Plot the critical points on the number line. Note that n critical points will divide the number line in (n+1) regions.
Step 5: In the right most region, the expression will be positive and in other regions it will be alternatively positive and negative. So, mark positive in the right most region and than mark alternatively positive and negative signs in the remaining regions.
Step 6: Obtain the solution set of the Inequation by selecting the appropriate region in step 5.
EXERCISE -1
1) 4x³- 24x²+ 44x - 24> 0. (1,2) U (3,∞)
2) 1/(x+1) - 4/(2+ x)² > 0, x≠ -1, -2. (-1,0) U (0,∞)
3) (1- x²)/(5x - 6 - x²) < 0. x∈ (-1,1) U (2,3)
4) (8x²+16x -51)/(2x²+5x-12) > 3. x∈(- ∞-4) U(-3, 3/2)U (5/2,∞)
5) (x² -2x +5)/(3x² -2x- 5) > 1/2. x∈(-5, -1)U (5/3,3)
6) (x² -2x +24)/(x²-3x+4) ≤ 4. x∈(- ∞-2/3) U[4,∞)
7) (x² -4x +7)/(x²-7x+12) ≤ 2/3. x∈[-3,1] ∈ (3,4)
8) Show that the function f(x)= 2x +3 is strictly increasing function on R.
9) Show that the function f(x)= -3x +12 is strictly decreasing function on R.
10) Show that the function f(x)= 7x - 3 is strictly increasing function on R.
11) Show that the function f(x)= -ax + b, where a, b are constants and a > 0 is an increasing function on R.
12) Show that the function f(x)= ax +b, where a, b are constant and a< 0 is a decreasing function on R.
13) Show that the function f(x)= 1/x is a decreasing function on (0,∞)
14) Show that the function f(x)= 1/(1+ x²) decreasing in the interval [0,∞) and increasing in the interval (-∞, 0].
15) Show that f(x)= 1/(1+ x²) is neither increasing nor decreasing on R.
16) Show that the function f(x)= x² is strictly increasing function on [0, ∞).
17) Show that the function f(x)= x² is a strictly decreasing function on (- ∞, 0].
18) Show that the function f(x)= x² is neither strictly increasing nor decreasing on R.
19) Show that the function f(x)= aˣ, a > 1 is strictly increasing on R.
20) Show that the function f(x)= aˣ, 0 <a < 1 is strictly decreasing on R.
21) Show that the function f(x)= logₑx is increasing on (0, ∞).
22) Show that the function f(x)= logₐx is increasing on (0, ∞) if a> 1 and decreasing on (0, ∞), if 0 < a < 1.
23) Show that the function f(x)= |x| is
a) strictly increasing (0, ∞)
b) strictly decreasing in (-∞, 0).
*********
• For sin - 1st +ve 2nd & 3rd -ve
In order to find the interval in which a given function is increasing or decreasing, use the following steps
Step 1: Obtain the function and put it equal to f(x).
Step 2: Find f'(x).
Step 3: Put f'(x)>0 and solve this Inequation.
For the values of x obtained in step 2 f(x) is increasing and for the remaining points in its domain it is decreasing.
**************************
EXERCISE -2
A) Find the intervals of the following is increasing or decreasing:
1) 10-6x-2x². (-∞,-3/2), (-3/2,∞)
2) 6 - 9x- x². (-∞,-9/2), (-9/2,∞)
3) x²+ 2x-5. (-1,∞),(-∞,-1)
4) f(x)= - x²-2x+25. (-∞,-1), (-1,∞)
5) f(x)= x³-6x²-36x+2. (-∞,-2) U (6,∞), (-2,-6)
6) 2x³ -9x²+12x+15. (-∞,-1) U (2,∞), (1,2)
7) f(x)= 2x³+9x²+12x+20. (-∞,-2) U (-1,∞), (-2,-1)
8) f(x)= 2x³+9x²+12x+20. (-∞,-2) U (-1,∞), (-2,-1)
9) f(x)= 2x³-15x²+36x+1. (-∞,2) U (3,∞), (2,3)
10) f(x)= 2x³- 24x+107. (-∞,-2) U (2,∞), (-2,2)
11) f(x)= 2x³-24x+7. (-∞,-2) U (2,∞), (-2,2)
12) 5+ 36x + 3x²-2x³. (-2,3), (-∞,-2) U (3,∞),
13) 8+ 36x + 3x²-2x³. (-2,3), (-∞,-2) U (3,∞),
14) 6+ 12x + 3x²-2x³. (-1,2), (-∞,-1) U (2,∞),
15) f(x)= x⁴ - x³/3. (1/4,∞), (-∞,1/4)
16) f(x)= x⁴- 8x³+ 22x²-24x+21. (1,2) U(3,∞), (1,2) U(3,∞)
17) 3x⁴/10 - 4x³/5 -3x²+ 36x/5 +11. (-2,1) U(3,∞), (-∞,-2)U(1,3)
18) x⁴/4 +2x³/3 -5x²/2-6x +7. (-3,-1) U(2,∞), (-∞,-3)U(-1,2)
19) {x(x-2)}². (0,1)U(1,∞)
20) f(x)= log(1+x) - 2x/(2+x) (-1,∞)
21) f(x)= x/2 + 2/x. (-∞,-2) U(2,∞), (-2,0)U(0,2)
22) f(x)= x³+ 1/x³, x≠0 (-∞,-1) U(-1,∞), (-1,0)U(0,1)
23) f(x)= (4x²+1)/x. (-∞,-1/2), (1/3,∞)
24) f(x)= (x +1)³(x -3)³. (1,3) U(3,∞), (-∞, -1) U (-1,1)
25) f(x)= (x -1)³(x -2)². (-∞,1) U(1, 8/5) U(2,∞), (8/5,2)
EXERCISE - 3
A) prove that the function is increasing in R:
1) f(x)= x³- 3x²+ 3x -100.
2) f(x)= x³-15x²+75x -50.
3) f(x)= x³- 6x²+ 12x - 18.
4) f(x)= x⁹+ 4x⁷+ 11.
5) f(x)= e²ˣ
6) f(x)= (x -1)eˣ + 1 for all x> 0.
7) f(x)=Tan⁻¹(sinx + cosx) at (0,π/4)
8) f(x)= 4sinx/(2+ cosx) at (0,π/2).
9) x - sinx.
B) Prove that the function is decreasing in
1) f(x)= e¹⁾ˣ, x ≠ 0 for all x≠ 0.
2) f(x)= kogₐx, 0 < a < 1 for all x > 0.
3) Prove that the function is neither increasing nor decreasing
f(x)= x²- x+1 on (0,1)
EXERCISE - C
1) Find the least value of a such that the function :
a) x²+ ax +1 is increasing on [1,2]. -2
b) x³- ax is an increasing on R. a≤0
c) sinx - ax + c is a decreasing on R. a≥1
d) x + cosx - a is an increasing on R.
2) 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 in the curve y = x²- 6x + 9 where the normal is parallel to the line y= x +5. Inc:(3,∞), dec: (-∞,3); (5/2,1/4)
3) Show that f(x)= sinx is increasing on (0,π/2) and decreasing on (π/2,π) and neither increasing nor decreasing in (0,π).
4) Separate [0,π/2] into subinterval in which f(x)= sin 3x is increasing or decreasing. (0,π/6), ((π/6,π/2)
5) Find the interval in which the function f given by f(x)= sinx+ cosx, 0≤ x≤ 2π is increasing or decreasing. (0,π/4) U (5π/4,2π), (π/4,5π/4)
6) Find the interval in which the function f given by f(x)= (4sinx-2x - x)/(2+ cosx) 0≤ x≤ 2π is increasing or decreasing. (0,π/2) U (3π/2,2π), (π/2,3π/2)
7) Show that f(x)= log sinx is increasing on (0,π/2) and decreasing on (π/2,π).
8) Determine the values of x for which it is increasing or decreasing
a) f(x)= xˣ, x>0. (1/e, ∞), (0,1)e)
b) x/log x. (e,∞), ((0,e)-{1}
c) f(x)= e¹⁾ˣ, x≠0 is a decreasing function for all x≠0
d) cos²x is a decreasing on (0,π/2)
No comments:
Post a Comment