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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

Let f (x) = 3x⁴ – 8x³ + 12x² – 48x + 25
f ' (x) = 12x³ – 24x² + 24x – 48
f ' (x) = 0 ⇒ 12x³ – 24x² + 24x – 48 = 0 ⇒ x³– 2x²+ 2x – 4 = 0
⇒ (x – 2) (x² + 2) = 0 ⇒ x = 2 , ± i $square root of 2$ .
Now x = 2 is the only real value in [0, 3]
f (0) = 0 – 0 + 0 – 0 + 25 = 25
f (2) = 48 – 64 + 48 – 96 + 25 = – 39
f (3) = 243 – 216 + 108 – 144 + 25 = 16
∴ max. value = 25 and min. value = – 39.

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.

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Short Answer Type

266.

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

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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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