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Application of Derivatives

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Long Answer Type

261.

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

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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

0

1

3

3

Let space space straight y space equals space fraction numerator 1 minus straight x plus straight x squared over denominator 1 plus straight x plus straight x squared end fraction

space dy over dx space equals space fraction numerator left parenthesis 1 plus straight x plus straight x squared right parenthesis. space begin display style straight d over dx end style left parenthesis 1 minus straight x plus straight x squared right parenthesis space minus space left parenthesis 1 plus straight x plus straight x squared right parenthesis. space begin display style straight d over dx end style left parenthesis 1 plus straight x plus straight x squared right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction

equals space fraction numerator left parenthesis 1 plus straight x plus straight x squared right parenthesis thin space left parenthesis negative 1 plus 2 straight x right parenthesis minus left parenthesis 1 minus straight x plus straight x squared right parenthesis thin space left parenthesis 1 plus 2 straight x right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction equals fraction numerator negative 1 plus 2 straight x minus straight x plus 2 straight x squared minus straight x squared plus 2 straight x cubed minus 1 minus 2 straight x plus straight x plus 2 straight x squared minus straight x squared minus 2 straight x cubed over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction equals space fraction numerator negative 2 plus 2 straight x squared over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis end fraction space equals space fraction numerator 2 left parenthesis straight x squared minus 1 right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction space equals space fraction numerator 2 left parenthesis straight x minus 1 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis squared end fraction

dy over dx space equals 0 space space space space rightwards double arrow space space space space fraction numerator 2 left parenthesis straight x squared minus 1 right parenthesis over denominator left parenthesis 1 plus straight x plus straight x squared right parenthesis end fraction space equals space 0 space space space space rightwards double arrow space space space straight x squared minus 1 space equals space 0 space space space space rightwards double arrow space space space straight x space equals space minus 1 comma space space 1 When space straight x less than negative 1 space slightly comma space space space dy over dx space equals space plus ve When space straight x space greater than negative 1 space slightly comma space dy over dx space equals space minus ve therefore space space space space space at space straight x space equals space minus 1 comma space space space dy over dx space changes space from space plus ve space to space minus ve therefore space space space space space space straight y space is space max comma space when space straight x space equals space minus 1 we space are space not space increased space in space this space case. When space straight x less than 1 space slightly comma space dy over dx space equals negative ve When space straight x greater than 1 space sightl comma space dy over straight d equals plus ve therefore space space at space straight x space equals space 1 comma space space dy over dx space changes space from space minus ve space to space plus ve therefore space space space space space space space straight y space is space minimum space when space straight x space equals space 1 and space minimum space value space space equals space fraction numerator 1 minus 1 plus 1 over denominator 1 plus 1 plus 1 end fraction space equals 1 third therefore space space space left parenthesis straight D right parenthesis space is space correct space answer.

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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