Wednesday, 4 May 2016

[Q] Trigonometry Tricks - 4 [Maximum and Minimum Values]

Few points to remember:

  1. sinθ and cosθ both have "1" as their maximum value and "-1" as their minimum value. Hence the values of sinx, sin2x, cosx, cos3x, etc. lie between -1 and 1.
  2. For sin2x and cos2x
    • Minimum value = 0
    • Maximum value = 1
  3. For sinxcosx
    • Minimum value = -1/2
    • Maximum value = 1/2
  4. Minimum value of (sin θ cos θ)n = (-1/2)n



Some SSC CGL questions :

      Q. 1) What is the least value of 2sin2θ + 3cos2θ
       (A) 1            (B) 2             (C) 3            (D) 0

      Since 2 is less than 3
      Minimum value = 2
      Answer: (B)

       

     Hence the least value of 4sec2θ + 9cosec2θ = 13 + 12 = 25
     Answer: (C)

      Q. 4) The maximum of 3sinx - 4cosx is
       (A) -1               (B) 5                (C) 7               (D) 9
       Maximum value = √(3^2 + 4^2) = √25 = 5
       Answer: (B)

Sometimes they ask the minimum/maximum value of a function. In CGL, that function would always be quadratic (ax2 + bx + c). Please note that a quadratic function can't have both maximum and minimum values. 

  • If "a" is positive, then the quadratic function will only have a minimum value. The maximum value would be infinite.
  • If "a" is negative, then the quadratic function will only have a maximum value. The minimum value would be infinite.

But the process to find both minimum and maximum values is same. Hence you shouldn't be worried about the words "maximum" or "minimum". When finding the minimum/maximum value of a function, we use the concept of "differentiation". Although differentiation is a wide topic in itself, but for CGL purpose, we only have to learn the basics.

Just remember following things:
  1. Differentiation of axn = a*n*xn-1Hence differentiation of 4x3 = 12x2 and differentiation of 3x2 = 6x
  2. Differentiation of ax = a. Hence differentiation of 4x is 4.
  3. Differentiation of any constant is zero. Hence differentiation of 5 is 0.
  4. If you have to differentiate ax2 + bx + c, just differentiate each of its term separately and add the result.
  5. Process of finding the minimum value of a function :
    1. Differentiate the function
    2. What ever result you get, equate it with zero
    3. Find the value of x
    4. Put this value of x in the original function to get the minimum value.

Let us solve a CGL question:

Q. 5) Find the minimum value of (x - 2)(x - 9)
(A) -11/4             (B) 49/4             (C) 0            (D) -49/4


(x - 2)(x - 9) = x^2 - 11x + 18
Now we will differentiate (x^2 - 11x + 18)
Note that in this question, the value of a is positive, i.e., +1 and hence the question has asked you the "minimum" value. Had the value of "a" been negative (like -1), they would have asked you the "maximum" value.

Hence, differentiation of (x^2 - 11x + 18) = 2x - 11
Now 2x - 11 = 0
x = 11/2
Put x = 11/2 in (x^2 - 11x + 18) to get the minimum value
Minimum value of x^2 - 11x + 18 = (11/2)^2 - 11*(11/2) + 18 = -49/4
Answer: (D)

If you have any doubt in this article, please drop a comment...
Keep reading :)


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21 comments:

  1. Another wonderful article.its fantabulous sir.Please post some reasoning hacks and general hacks for cgl like eliminating options,option substitution etc

    ReplyDelete
  2. This topic of trigo was troubling me for quite sometime. This article explains everything with utmost clarity. Amazing work sir. Thank You so much.

    ReplyDelete
  3. Sir please post hack for English too...
    Bcs this being a rank decided section of cgl
    Thanx in advance

    ReplyDelete
  4. Thanx for ur genuine effort sir.
    u hv given explaination for finding min value of quadratic eqn. Plse provide us the method to find its max value too.

    ReplyDelete
  5. Thanx for ur genuine effort sir.
    u hv given explaination for finding min value of quadratic eqn. Plse provide us the method to find its max value too.

    ReplyDelete
    Replies
    1. The process to find the minimum and maximum values is same. I have added an explanation in the article. Please read the quadratic function part again.

      Delete
  6. Sir ,
    I have filled general as my category in ssc cpo exam
    But i have obc certificate. Can i make changes after exam or now? Pls tell sir ... I have delhi obc certificate ,i was unaware that in delhi police ,i can use ..

    ReplyDelete
    Replies
    1. No. You cant make any changes after exams or now.

      Delete
  7. Thankyou so much sir
    Your blogs are really helpful

    ReplyDelete
  8. Thankyou so much sir
    Your blogs are really helpful

    ReplyDelete
  9. Thanks a lot sir. Sir please write about "average".

    ReplyDelete
  10. Thanks a lot sir. Sir please write about "average".

    ReplyDelete
  11. Sir ,,aapne kahi se bhi coaching nahi li thi kya????
    Every time this question is attack on my mind

    ReplyDelete
  12. Thank You so much Sir.... It has been a wonderful reading experience for CGL....Can you plz try to put up some tricks related to English Section as well?

    ReplyDelete
    Replies
    1. Chill...still 4 months are left :)

      Delete
  13. Hats off sir
    Itne logo ki dil se duaaye h
    Realy padhte2 aansu aa gye
    Thanks a lot sir

    ReplyDelete
  14. Hats off sir
    Itne logo ki dil se duaaye h
    Realy padhte2 aansu aa gye
    Thanks a lot sir

    ReplyDelete
  15. I must tell you Sir that the efforts you are putting in for the aspirants are simply amazing.
    You owe us a lot.
    If I crack CGL 2017,I would definitely want to host you for a treat. :) Thanks

    ReplyDelete