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Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Concept: Increasing and Decreasing Functions
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Concept: Maxima and Minima
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Concept: Increasing and Decreasing Functions
Find the approximate value of cos (60° 30').
(Given: 1° = 0.0175c, sin 60° = 0.8660)
Concept: Approximations
Find the approximate value of log10 (1016), given that log10e = 0⋅4343.
Concept: Approximations
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Concept: Maxima and Minima
Find the equation of the tangent to the curve at the point on it.
y = x2 + 2ex + 2 at (0, 4)
Concept: Applications of Derivatives in Geometry
A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?
Concept: Derivatives as a Rate Measure
Verify Lagrange’s mean value theorem for the following function:
f(x) = log x, on [1, e]
Concept: Lagrange's Mean Value Theorem (LMVT)
Divide the number 20 into two parts such that sum of their squares is minimum.
Concept: Maxima and Minima
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Concept: Maxima and Minima
Choose the correct option from the given alternatives:
Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.
Concept: Applications of Derivatives in Geometry
Choose the correct option from the given alternatives :
If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x`, then ______.
Concept: Applications of Derivatives in Geometry
The approximate value of tan (44°30'), given that 1° = 0.0175c, is ______.
Concept: Approximations
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Concept: Maxima and Minima
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Concept: Maxima and Minima
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Concept: Maxima and Minima
The slope of the normal to the curve y = x2 + 2ex + 2 at (0, 4) is ______.
Concept: Applications of Derivatives in Geometry
If the tangent at (1, 1) on y2 = x(2 − x)2 meets the curve again at P, then P is
Concept: Applications of Derivatives in Geometry
The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. The time when the acceleration is 14 ft/sec2, is
Concept: Derivatives as a Rate Measure
