MAXIMUM AND MINIMUM
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1) Find maximum and minimum value.
a) 3x²+ 6x +8, at x=-1 min=5
b) 4x² - 4x +4. Min=3
c) -(x-1)² +2. Max= 2
d) 16x² - 16x +28. Min= 24
e) 2x³ -15x²+36x+10. 38, 37
f) 2x³ - 21x²+36x - 20. -3, -128
g) x³ - 6x²+9x+15. 19,15
h)
i) 7x⁵- 15x⁴+10x². 21/16,0
j) x²(x-1)³. 0, -108/3125
k) (x-1)(x+2)². 0,-4
l) 2/x - 2/x², x> 0. Maxv= 1/2
m) -3x⁴/4 - 7x³ -45x²/2+105. 73.75 and 105, 57.75.
n) 1/(x²+1). Max- 1
o) x/{(x-1)(x-4)}. -1, -2
p) x³ - 3x +1. 3, -1
q) √(a+x) + √(a-x). √(2a)
r) (x²-7x+6)/(x-10). 16
s) x + 1/(x+1). -3, 1
t) (x+3)²(x-2)². 625/16, 0
u) sin 2x +5. Max=6
v) Sin2x ; 0<x<π,. 1, -1
w) sin2x -x. -√3/2 +π/4,√3/2 +π/4,
x) Sinx + (cos2x)/2, 0≤x≤π/2. 3/4, 1/2
y) 2 sinx - x; -π/2≤x≤π/2,. √3-π/3, √3+π/3
z) x+2 cos x ; 0 <x<π. At x=π/6, max- π/6 + √3
a') Sin⁴x + Cos⁴x; 0<x<π/2. Min= 1/2
b') 2sin x + cos 2x ;0<x<2π. 5π/6, 3π/2
c') a sinx+ b cosx. √(a²+b²), -√(a²+b²)
d') xeˣ. Minv= -1/e
e') sin x - cos x at 0 <x < 2π. √2,-√2
f') sinx + cosx ; 0< x<π/2. Max=√2
g') Sin x(1+cos x) at 0 < x < π/2. 3√3/4,
h') y= 4/(x+2) + x. -6, 2
i') y= tanx - 2x. -1-3π/2, 1-π/2
2) PROVE:
a) Show that (1/x)ˣ is e¹⁾ᵉ.
b) Show that the maximum value of x³ + 1/x³ is less than minimum its value.
c) The function y= a logx + bx² + x has extreme values at x= 1 and x= 2. Find a and b. -2/3, -1/6
d) (logx)/x has a maximum value at x= e.
e) If f(x)= x³ + ax² + bx + c has a maximum at x=-1 and minimum at x= 3. Determine a,b,c. -3,-9, c belongs to R.
3) Find Maximum and Minimum Value of the following:
a) f(x)= 2x³- 24x +107 at [1,3]. 89, 75
b) f(x)= 4x - x²/2 in[-2, 4.5]. 8, -10
c) x⁵⁰ - x²⁰ [0,1] . 0, -3/5 ³√(2/5)
d) (1/2 - x)²+ x³ in [-2,2.5]. 157/8, -7/4
e) 3x⁴ - 8x³+ 12x²- 48x + 1 on [1,4]. -59, 257
f) 3x⁴ - 8x³+ 12x²- 48x + 25 on [0,3]. 25, -39
g) (x-1)²+3 in [-3,1]. 19, 3
g) sinx at [π,2π]. 0, -1
i) Sinx +1/2 cos2x, [0,π/2]. 3/4,1/2
j) x+ sin 2x at [0,2π]. 2π, 0
k) Sinx+ cos x in [0,π]. √2, -1
4) Show that Sinx(1+ cosx) is maximum at x=π/3 at [0,π].
5) It is given that at x=1, the function x⁴ - 62x² + ax +9 attains it's Maximum value on the interval [0,2]. Find the value of a. 120
6) Find the maximum value of 2x³- 24x + 107 in the interval [1,3]. Find the Maximum value of the same function in [-3,-1]. Maxv= 89 at x=3, max v= 139 at x=-2.
7) Find the local maxima and minima of:
a) x³ -6x²+ 9x -8. 1, 3
b) (x-1)³(x+1)².
Continue......
Type--2:
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1) Find two positive numbers x and y such that x+y= 60 and xy³ is Maximum. 15,45
2) Find two positive numbers x and y such that their sum is 35 and product x²y⁵ is Maximum. 10, 25
3) Amongst all pairs of positive numbers with product 256, find those whose sum is the least. 16
4) Determine two positive real Number whose sum is 14 and the sum of whose squares is least. 7,7
5) Find two positive numbers whose sum is 15 and the sum of whose squares is Minmum. 15/2, 15/2
6) Divide 64 into two parts such that the sum of the cubes of two parts is minimum. 32,32
7) The sum of two positive numbers is 30. Find the numbers if the sum of their squares is to be minimum. 15,15
8) The product of two positive numbers is 25. Find the numbers if their sum is to be minimum. 5,5
9) The difference of two numbers is 100. The square of the larger one exceeds five times the square of the smaller one by an amount which is maximum. Find the numbers. 25,125
10) How should we choose two numbers, each greater than or equal to -2, whose sum is ½ so that the sum of the first and the cube of the second is minimum. (1/2-1/√3), 1/√3
11) Show that among all positive numbers x and y with x²+y²= r², the sum x+y is largest when x=y=r/√2.
12) Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum. 6,9
13) Find the maximum slope of the curve y= -x³+ 3x²+ 2x -27. 5 at(1,-23)
Type -3
1) A wire of length 8 cm is to form a rectangle. Find the dimensions of the rectangle.
) Find the dimensions of the rectangle of area 96cm³ whose perimeter is the least. Find also it's perimeter.
) The perimeter of a rectangle is 100cm. If the area is minimum, find the lengths of its sides. 25,25
) Show that, of all rectangle of given perimeter, the square has the greatest area.
) Prove, of all rectangles of a given area, the square has the least perimeter.
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17) A figure consists of a semicircle with a rectangle on its diameter given that the perimeter of the figure is 20m, find the dimensions in order that its area may be maximum.
18) A wire of length 36m is to be cut into two pieces. One of the piece is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum.
19) Find the radius of a closed right circular cylinder of volume 100cm³ which has minimum total surface area.
20) A rectangle is inscribed in a semi-circle of radius r with one of its sides on the diameter of the semi-circle. Find the dimensions of the rectangle so that its area is maximum.
22) An open tank with a square base of side x metres and vertical height h metres is to be construed so as to contain C cm³ of water. Show that the expenses on lining the inside of the tank with lead would be least if h= x/2.
VERY SHORT ANSWER QUESTIONS
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EXERCISE - 2
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1) Write necessary condition for a point x= c to be an extreme point of the function f(x).
2) Write sufficient conditions for a point x= c to be a point of maximum.
3) If f(x) attains a minimum at x= c, then write the values of f'(c) and f"(c).
4) Write the minimum value of f(x)= x+ 1/x, x> 0.
5) Write the maximum value of f(x)= x+ 1/x, x < 0.
6) write the point where f(x)= x log x attains minimum value.
7) Find the least value of f(x)= ax + b/x, where a> 0, b> 0 and x> 0.
8) write the minimum value of f(x)= xˣ.
9) Write the maximum value of f(x)= x¹⁾ˣ
10) Write the maximum value of f(x)=( logx)/x, if it exists.
Answer)
1) f'(c)= 0 2)f'(c)= 0 & f''(c)<0
3) f'(c)= 0 & f''(c)>0
4) 2 5) -2 6) (1/e, -1/e)
7) 2√(ab) 8) r⁻¹⁾ᵉ 9) e¹⁾ᵉ 10) 1/e
MULTIPLE CHOICE QUESTIONS
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EXERCISE -3
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1) The maximum value of x¹⁾ˣ, x> 0
A) e¹⁾ᵉ B) (1/e)ᵉ C) 1 D) n
2) If ax + b/x ≥ c for all x where a, b, > 0, then.
A) ab<c²/4 B) ab≥ c²/4 C) ab≥ c/4 D) n
3) The minimum value of x/(logx) is
A) e B) 1/e C) 1 D) none
4) For the function f(x)= x+ 1/x
A) x=1 is a point of a maximum
B) x=-1 is a point of a minimum
C) maximum value> minimum value
D) maximum value < minimum value
5) Let f(x)= x³+ 3x²-9x+2 then, f(x) has
A) a maximum at x= 1
B) a minimum value at x= 1
C) neither a maximum nor a minimum at x=-3 D) none
6) the minimum value of f(x)= x⁴- x²-2x+6 is
A) 6 B) 4 C) 8 D) none
7) the number which exceeds its square by the greatest possible quantity is
A) 1/2 B) 1/4 C) 3/4 D) none
8) let f(x)= (x-a)²+(x -b)²+(x-c)². then f(x) has a minimum at x=
A)(a+b+c)/3 B) ³√(abc) C)3/(1/a+ 1/b+ 1/c) D) none
9) The sum of two non zero numbers is 8. The minimum value of the sum of their reciprocals is
A) 1/4 B) 1/2 C) 1/8 D) n
10) the functions f(x)= ⁵ᵣ₌₁∑(x-r)² assume minimum value at x=
A) 5 B)5/2 C) 3 D) 2
11) at x= 5π/6, f(x)= 2 sin 3x + 3
A) 0 B) maximum C) minimum D) n
12) if x lies in the interval [0,1], then the least value of x²+x+1 is
A) 3 B)3/4 C) 1 D) none
13) the least value of the function f(x)= x³ - 18x² +96x in the interval [0,9] is
A) 126 B) 135 C) 160 D) 0
14) the maximum value of f(x)= x/(4 - x +x²) on [-1,1] is
A) -1/4 B) -1/3 C) 1/6 D) 1/5
15) the point on the curve y²= 4x which is nearest to the point (2,1) is
A) (1,2√2) B) (1,2) C) (1,-2) D) (-2,1)
16) if x+y= 8, then the maximum value of xy is
A) 8 B) 16 C) 20 D) 24
17) the least and the greatest value of f(x)= x³ - 6x² +9x in [0,6], are
A) 3,4 B) 0,6 C) 0,3 D) 3,6
18) f(x)= sinx + √3 cosx is maximum when x=
A) π/3 B) π/4 C) π/6 D) 0
19) If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is
A) 3/4. B) 1/3 C) 1/4 D) 2/3
20) the minimum value of (x² + 250/x) is
A) 75 B) 50 C) 25 D) 55
21) If f(x)= x + 1/x, x> 0, then its greatest value is
A) -2 B) 0 C) 3 D) none
22) If f(x)= 1/(4x² +2x+1), then its maximum value is
A) 4/3 B) 2/3 C) 1 D) 3/4
23) let x,y be two variables and x> 0, xy= 1, then minimum value of x+y is
A)1 B) 2 C) 5/2 D) 10/3
24) if f(x)= x + 1/x, x> 0, then its greatest value is
A) -2. B) 0 C) 3 D) none
25) the function f(x)= 2x³ - 15x² +36x +4 is maximum at x=
A)3 B) 0 C) 4 D) 2
26) the maximum value of f(x)=x/(x²+x+4) on [-1,1] is
A) -1/4 B) -1/3 C) 1/6 D) 1/5
27) Let f(x)= 2x³ - 3x² -12x +5 on [-2,4] the relative maximum occurs at x=
A) -2 B) -1 C) 2 D) 4
28) the minimum value of x log x is equals to
A) e B) 1/e C) -1/e D) 2/e E) -e
29) the minimum value of the function f(x)= 2x³ - 21x² +36x-20 is
A) -128 B) -126 C) -120 D) none
30) f(x)= 3cos²x+ 2sinx +1, 0≤ x≤ 2π/3 is
A) minimum at x=π/2
B) maximum at x= sin⁻¹(1/√3)
C) minimum at x=π/6
D) maximum at x= sin⁻¹(1/6)
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