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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
A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______
Concept: Rolle's Theorem
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Concept: Increasing and Decreasing Functions
Find points on the curve given by y = x3 − 6x2 + x + 3, where the tangents are parallel to the line y = x + 5.
Concept: Applications of Derivatives in Geometry
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
Concept: Maxima and Minima
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
Concept: Maxima and Minima
The maximum value of the function f(x) = `logx/x` is ______.
Concept: Maxima and Minima
Find the equation of tangent to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`.
Concept: Applications of Derivatives in Geometry
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Concept: Increasing and Decreasing Functions
Find the approximate value of sin (30° 30′). Give that 1° = 0.0175c and cos 30° = 0.866
Concept: Approximations
