Find the intervals in which the function f is given by (i) incr

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 Multiple Choice QuestionsShort Answer Type

181.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
2x3 – 8x2 + 10x + 5

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182.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
2x3 – 6x2 – 48x + 17

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183.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f (x) = 2x– 9x2 + 12x + 30

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184.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f (x) = 2x3 – 3x2 – 36x + 7

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185.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f(x) = 2x3 – 21x2 + 36x – 40 

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186.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
4x3 – 6x2 – 72x + 30

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187.

Find the intervals in which the following functions are strictly increasing or strictly decreasing:
– 2x3 – 9x2 – 12x + 1

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188.

Find the intervals in which the function
straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 straight x squared plus 1 over denominator straight x end fraction comma space space straight x not equal to 0, is
(a) increasing (b) decreasing.

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 Multiple Choice QuestionsLong Answer Type

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189.

Find the intervals in which the function f is given by 
straight f left parenthesis straight x right parenthesis space equals space straight x cubed plus 1 over straight x cubed comma space space space straight x not equal to 0 space space is space
(i) increasing    (ii) decreasing


Here straight f left parenthesis straight x right parenthesis space equals space straight x cubed plus 1 over straight x cubed
therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 3 straight x squared minus 3 over straight x to the power of 4 space equals space fraction numerator 3 left parenthesis straight x to the power of 6 minus 1 right parenthesis over denominator straight x to the power of 4 end fraction space equals space 3 over straight x to the power of 4 left square bracket left parenthesis straight x squared right parenthesis cubed space minus space left parenthesis 1 right parenthesis cubed right square bracket
space space space space space space space space space space space space space space space space space equals 3 over straight x to the power of 4 left square bracket left parenthesis straight x squared minus 1 right parenthesis space left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis right square bracket space equals space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction left parenthesis straight x squared minus 1 right parenthesis
(i) For straight f left parenthesis straight x right parenthesis to be increasing
                     straight f apostrophe left parenthesis straight x right parenthesis space greater than space 0 space space space space space space space rightwards double arrow space space space space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction left parenthesis straight x squared minus 1 right parenthesis space greater than space 0
     rightwards double arrow space space space straight x squared minus 1 space greater than 0                                                                     open square brackets because space space space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction greater than 0 close square brackets
rightwards double arrow space space space space space straight x squared minus 1 space space space rightwards double arrow space space space open vertical bar straight x close vertical bar squared space greater than space left parenthesis 1 right parenthesis squared space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar space greater than space 1
rightwards double arrow space space space either space straight x space less than space minus 1 space space space or space space space straight x greater than 1

(ii) For f(x) to be decreasing,
   straight f apostrophe left parenthesis straight x right parenthesis space less than space 0 space space space space space rightwards double arrow space space space space fraction numerator 3 left parenthesis straight x to the power of 4 plus straight x squared plus 1 right parenthesis over denominator straight x to the power of 4 end fraction left parenthesis straight x squared minus 1 right parenthesis thin space less than space 0
rightwards double arrow space space space space straight x squared minus 1 space less than space 0 space space space space rightwards double arrow space space space straight x squared less than 1 space space space space space rightwards double arrow space space space space open vertical bar straight x close vertical bar squared space less than space left parenthesis 1 right parenthesis squared space space space rightwards double arrow space space space open vertical bar straight x close vertical bar space less than space 1
rightwards double arrow space space space space space space minus 1 less than straight x less than 1.

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190.

Determine for which values of x, the function f (x) = x4 – 2x2 is increasing or decreasing.

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