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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 $left square bracket 0 comma space 2 straight pi right square bracket$ .

75 Views

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 $fraction numerator log space straight x over denominator straight x end fraction space in space 0 less than straight x less than infinity$ .

Let space straight y space equals space fraction numerator log space straight x over denominator straight x end fraction therefore space space space space dy over dx space equals space fraction numerator straight x. space begin display style 1 over straight x end style minus log space straight x. space 1 over denominator straight x squared end fraction space equals space fraction numerator 1 minus log space straight x over denominator straight x squared end fraction space space space space space space space space dy over dx space equals space 0 space gives space us space fraction numerator 1 minus logx over denominator straight x squared end fraction space equals space 0 rightwards double arrow space space space 1 minus logx space equals space 0 space space space space space rightwards double arrow space space space space logx space equals space 1 space equals space loge space space space space rightwards double arrow space space straight x space equals space straight e space space space space space space space space space space space space space dy squared over dx squared space equals space fraction numerator straight x squared open parentheses negative begin display style 1 over straight x end style close parentheses minus left parenthesis 1 minus log space straight x right parenthesis. space 2 straight x over denominator left parenthesis straight x squared right parenthesis squared end fraction space equals space fraction numerator negative straight x minus 2 straight x plus 2 straight x space logx over denominator straight x to the power of 4 end fraction space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator negative 3 straight x plus 2 straight x space logx over denominator straight x to the power of 4 end fraction space equals space fraction numerator negative 3 plus 2 space log space straight x over denominator straight x cubed end fraction At space straight x space equals space straight e comma space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator negative 3 plus 2 space log space over denominator straight e cubed end fraction space equals space fraction numerator negative 3 plus 2 over denominator straight e cubed end fraction space equals space minus 1 over straight e cubed less than 0 space space space space space space space space space

therefore space space space space straight y space is space maximum space when space straight x space equals space straight e space and space maximum space value space space equals space fraction numerator log space straight e over denominator straight e end fraction space equals space 1 over straight e

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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