What is the maximum value of the function sin x + cos x? from

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 Multiple Choice QuestionsLong Answer Type

251. Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
straight x cubed left parenthesis straight x minus 1 right parenthesis squared

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252. Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
straight x cubed left parenthesis 2 straight x minus 1 right parenthesis cubed


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253. Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
negative left parenthesis straight x minus 1 right parenthesis cubed space left parenthesis straight x plus 1 right parenthesis squared



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 Multiple Choice QuestionsShort Answer Type

254.

Find all the local maxima or minima of the function
f (x) = x– 12x.

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255. Find all the points of local maxima and local minima of the function f given by f (x) = x3 – 27 x + 3. Also, find local maximum and local minimum value of f.
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256. Find all points of local maxima and local minima of the function f given by f (x) = x3  3x + 3.
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257. Find all the points of local maxima and local minima of the function f given by f (x) = 2x3 – 6x2 + 6x + 5 .
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258.

Find local minimum value of the function f given by f(x) = 3+ | x |, x ∊ R.

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259.

Find local maximum and local minimum values of the function f given by
f (a) = 3x4 + 4x3 – 12x2 + 12

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260.

What is the maximum value of the function sin x + cos x?


Consider the interval  open square brackets 0 comma space 2 straight pi close square brackets
Let space space straight f left parenthesis straight x right parenthesis space equals space sinx space plus cosx comma space space space space space space therefore space space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space cosx minus sinx
space space
         straight f apostrophe space left parenthesis straight x right parenthesis space equals space 0 space space space space gives space cosx minus sinx space equals space 0 space space space space space rightwards double arrow space space space sinx space equals space space cosx
rightwards double arrow space space space tanx space equals 1 space space space space space space rightwards double arrow space space space space straight x space equals space straight pi over 4 comma space space space fraction numerator 5 straight pi over denominator 4 end fraction

     Now comma space space space straight f left parenthesis 0 right parenthesis space equals space sin space 0 space plus space cos space space 0 space equals space 0 space plus space 1 space equals space 1
               straight f open parentheses straight pi over 4 close parentheses space equals space sin space straight pi over 4 plus space cos space straight pi over 4 space equals space fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction space equals space fraction numerator 2 over denominator square root of 2 end fraction space equals space square root of 2
straight f open parentheses fraction numerator 5 straight pi over denominator 4 end fraction close parentheses space equals space sin space space fraction numerator 5 straight pi over denominator 4 end fraction plus cos space fraction numerator 5 straight pi over denominator 4 end fraction space equals space sin space open parentheses straight pi plus straight pi over 4 close parentheses plus space cos space open parentheses straight pi plus straight pi over 4 close parentheses equals space space minus sin space straight pi over 4 minus cos space straight pi over 4
space space space space space space space space space space space space space space space equals negative fraction numerator 1 over denominator square root of 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction space equals space minus fraction numerator 2 over denominator square root of 2 end fraction space equals space minus square root of 2
space space space straight f left parenthesis 2 straight pi right parenthesis space equals space sin space 2 straight pi space plus space cos space 2 straight pi space equals space 0 plus 1 space equals space 1
therefore space space space space maximum space value space space equals space square root of 2

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