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Maximums and Minimums:

A function f(x) has relative maxima at x = a if f(a) is greater than the other values of f(x) or at this point the value of the function is maximum in its surroundings.

And if a function f(x) has relative minima at x = b if f(b) is lesser than the other values of f(x) or at this point the value of the function is minimum in its surroundings.

The point of local maxima or local minima is known as local extremum and the function is differentiable at x= a, and  at this point f’(a) = 0.

Maximums and Minimums Sample Questions:

Question 1: Find the minimum value of the function  |x+2|-1.

(a) 2

(b) 1

(c) 0

(d) -1

Explanation: |x+2|0

Subtracting 1 from both the sides

Question 2:  For f(x) =  |sin 4x +3| Which option is correct?

(a) min value = 4

(b) max value = 2

(c) min value =0

(d) min value = 2

Explanation: f(x) = |sin4x +3|

-1 sin4x 1

Question 3: Which of the following does not have maxima or minima?

(a) 4a +2x^2

(b) e^x

(c) t^3+2t^2+1

(d) 3sinx + cosx

Explanation: f(x) = e^x

f’(x) = e^x

Question 4: For real value of x,  the minimum value of x^2 – 6x +7

(a) -1

(b) 0

(c) -2

(d) 2

Explanation: f(x) = x^2 – 6x +7

f’(x) = 2x – 6

Question 5:  For f(x) = 2x^3 -3x^2-12x +4, it has

(a) two points of local maxima

(b) two points of local minima

(c) one maxima and one minima

(d) no maxima and no minima

Explanation: f(x) = 2x^3 -5x^2-12x+144

f’(x) = 6x^2 – 6x -12

Question 6: The function f(x) has the maximum value of sinx cosx

(a) 0.25

(b) 1/2

(c) 1

(d)

Explanation: We know that -1 sin 2x 1

-1 2 sinx cosx  1

Question 7: What is the absolute maximum value of y= x^3-3x+2 in [0,2]

(a) 4

(b) 6

(c) 0

(d) 2

Explanation: y = x^3- 3x+2

= 3x^2 -3

Question 8: For f(x,y) = 2x^2 +2xy –y^3

(a) only one stationary point at (0,0)

(b) two stationary point at (0,0) and (1/6,-1/3 )

(c) no stationary point

(d) two stationary point at (0,0) and (-1,1)

Explanation: f(x) = 2x^2 +2xy –y^3

= 4x +2y

Question 9: Find the maximum value of 2x^3 -9x^2 +12x +8 in the interval  [0,3]

(a) 2

(b) 1

(c) 0

(d) 6

Explanation: f(x) = 2x^3 -9x^2 +12x +8

f’(x) =  6x^2 -18x +12

Question 10: Harry buys a pizza and some bottles of soft drinks, the cost of one pizza is \$12.67 and that of soft drink is \$4.56 per bottle. If the total cost was \$22.76 then how many bottles of water did he buy?

(a) 0

(b) -9

(c) 14

(d) -13