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Long Answer Type

261.
Find the maximum and minimum value of x + sin 2 x on $left square bracket 0 comma space 2 straight pi right square bracket$ .

Let space straight f left parenthesis straight x right parenthesis space equals space straight x plus sin space 2 straight x

therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space space equals space 1 plus 2 space cos space 2 straight x space space space space space space space space space space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space space 1 plus 2 space cos 2 straight x space equals space 0 rightwards double arrow space space space space cos space 2 straight x space equals space minus 1 half space equals space minus cos space 60 degree space equals space minus cos space straight pi over 3 rightwards double arrow space space space cos space 2 straight x space space equals cos space open parentheses straight pi minus straight pi over 3 close parentheses comma space space space cos space open parentheses straight pi plus straight pi over 3 close parentheses comma space space space cos space open parentheses 3 straight pi minus straight pi over 3 close parentheses space cos space open parentheses 3 straight pi plus straight pi over 3 close parentheses rightwards double arrow space space space space space 2 straight x space equals space fraction numerator 2 straight pi over denominator 3 end fraction comma space space fraction numerator 4 straight pi over denominator 3 end fraction comma space fraction numerator 8 straight pi over denominator 3 end fraction comma space space fraction numerator 10 straight pi over denominator 3 end fraction rightwards double arrow space space space space space space space space space space space straight x space equals space straight pi over 3 comma space space fraction numerator 2 straight pi over denominator 3 end fraction comma space fraction numerator 4 straight pi over denominator 3 end fraction comma space space fraction numerator 5 straight pi over denominator 3 end fraction

straight f left parenthesis 0 right parenthesis space equals space 0 plus sin space 0 space equals space 0 plus 0 space equals space 0 straight f open parentheses straight pi over 3 close parentheses space equals space straight pi over 3 plus sin space fraction numerator 2 straight pi over denominator 3 end fraction space equals space straight pi over 3 plus fraction numerator square root of 3 over denominator 2 end fraction straight f open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space equals space fraction numerator 2 straight pi over denominator 3 end fraction plus sin space fraction numerator 4 straight pi over denominator 3 end fraction space equals space fraction numerator 2 straight pi over denominator 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction straight f open parentheses fraction numerator 4 straight pi over denominator 3 end fraction close parentheses space equals space fraction numerator 4 straight pi over denominator 3 end fraction plus sin space fraction numerator 8 straight pi over denominator 3 end fraction space equals space fraction numerator 4 straight pi over denominator 3 end fraction plus fraction numerator square root of 3 over denominator 2 end fraction straight f open parentheses fraction numerator 5 straight pi over denominator 3 end fraction close parentheses space equals space fraction numerator 5 straight pi over denominator 3 end fraction plus sin space fraction numerator 10 straight pi over denominator 3 end fraction space equals space fraction numerator 5 straight pi over denominator 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction

straight f left parenthesis 2 straight pi right parenthesis space equals space 2 space straight pi space plus space sin space 4 straight pi space equals space 2 straight pi plus 0 space equals space 2 straight pi

therefore space space space max space. value space space equals space 2 straight pi comma space space min. space value space equals space 0

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Short Answer Type

262.

Find both the maximum value and the minimum value of
3x⁴ – 8x³ + 12x² – 48x + 25

90 Views

263. It is given that at x = 1, function x⁴ – 62x² + ax + 9 attains its maximum value, on the interval [0, 2], Find the value of a.

88 Views

Long Answer Type

264.

straight y space equals space fraction numerator ax minus straight b over denominator left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x minus 4 right parenthesis end fraction

has a turning point at P(2, -1). Find the values of a and b and show that y is maximum at P.

74 Views

265. If y = a log | x | + bx² + x has its extremum values at x = – 1 and x = 2, find the values of a and b. Prove that these extrema are maximum.

74 Views

Short Answer Type

266.

Find the maximum and minimum values of $fraction numerator log space straight x over denominator straight x end fraction space in space 0 less than straight x less than infinity$ .

74 Views

Long Answer Type

267.

Find the shortest distance of the point (0, c) from the parabola y = x² , where 0 ≤ c ≤ 5.

160 Views

Short Answer Type

268. An Apache helicopter of enemy is flying along the curve given by y = x² + 7. A soldier, placed at (3. 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

139 Views

Multiple Choice Questions

269.

For all real values of x, the minimum value of $fraction numerator 1 minus straight x plus straight x squared over denominator 1 plus straight x plus straight x squared end fraction$ is

85 Views

270.

The maximum value of $open square brackets straight x left parenthesis straight x minus 1 right parenthesis space plus space 1 close square brackets to the power of 1 third end exponent comma space space space 0 space less or equal than space straight x space less or equal than space 1$ is

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