The following function has a local manima at which value of x
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Option: C Explanation : Click on Discuss to view users comments. |
If temperature field in a body varies according to the equation T(x, y)= x2 + 4xy, then direction of fastest variation in temperature at the point (1, 0) is given by
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Option: C Explanation :
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If f(x) = 3x4 -4x2 +5, then the interval for which f(x) satisfied all the condition of Rolle's Theorem is
| A. | [0, 2] |
| B. | [-1, 1] |
| C. | [-1,0] |
| D. | [1, 2] |
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Option: B Explanation :
Given function is, f(x) = 3x4 - 4x2 + 5
(i) f (x) is continuous in every real interval (ii) f ' (x) exist in any real interval (iii) f(-1) = 3(-1)4 - 4(1)2 + 5 = 4 f(1) = 3(1)4 - 4(1)2 + 5 = 4 f(-1) = f(1) Also f ' (c) = 12c3 - 8c = 0 ⇒ c = 0,  c ∈ [-1, 1]
Hence all the condition of Rolle's Theorem are satisfied in the interval [-1, 1]
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The triangle of maximum area inscribed in a circle of radius r is
| A. | a right angled triangle with hypotenuse measuring 2r |
| B. | an equilateral triangle |
| C. | an isosceles triangle of height r |
| D. | doesnot exist |
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Option: B Explanation :
Let ABC be a triangle inscribed in the circle with centre 0 and radius r. If area of this triangle is maximum, then vertex C should be at a maximum distance from the base AB i.e. , CD must be perpendicular to AB.
Hence ABC is an isosceles triangle.  If ∠ BCD = 0, where D is the mid.-point of BC, then ∠ BOD = 2ϑ
therefore, AB= 2BD = 2r sin 2ϑ
CD = CO + OD = r + r cos 2ϑ If S be the area of the triangle ABC, then S = 1/2 AB x CD = 1/2 x 2r sin 2 ϑ(r + r cos 2ϑ)
ds /dϑ = r2 [sin 2ϑ (-2 sin 2ϑ) + (1 + cos 2ϑ) (2 cos 2ϑ)]
= 2r2 [cos 22ϑ - sin2 2ϑ + cos 2ϑ] = 2r2 (cos 4ϑ + cos 2ϑ) For maximum and minimum, ds / dϑ = 0 cos 4ϑ + cos 2ϑ = 0 2 cos 3ϑ cos ϑ = 0
Hence either cos 3ϑ = 0, or cos ϑ = 0, which is impossible
if cos ϑ = 0, then ϑ = Π / 2
if cos 3ϑ = 0, then 3 ϑ = Π / 2 ϑ = Π / 6 ( a2s / dϑ 2) 6 = Π/6 is negative therefore S is maximum for ϑ = 1 / 6 * Π ∠ ACB = 2ϑ = 2( Π / 6 ) = Π / 3 = ∠ ABC = ∠ BAC Hence ABC is an equilateral triangle. Click on Discuss to view users comments. |
The greatest and least value of
f(x) = x4- 8x3 + 22x2 - 24x +1 in [0, 2] are
| A. | 0, 8 |
| B. | 0, -8 |
| C. | 1,8 |
| D. | 1, -8 |
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Option: D Explanation :
f(x) = x4- 8x3 + 22x2 - 24x +1 , f(0) = 1 Click on Discuss to view users comments. |
