A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the radius of the semicircular opening of the window to admit maximum light.
The point of local maxima for the function f(x) = sinx. cos x is
f(x) = sinx.cosx
f’(x) = -sin2x + cos2x
-sin2x + cos2x = 0
sin2x = cos2x
tan2x = 1
tanx = +-1
for tan x = 1, x = π/4
for tan x = -1, x = 3π/4
At x = π/4, f(π/4) = ½
At x = 3π/4, f(3π/4) = -½
At π/4 is the local maxima.
A real number x when added to its reciprocal give minimum value to the sum when x is
The sum of two positive numbers is 20. Find the numbers if their product is maximum
The function f(x) = log x
f(x) = log x
f’(x) = 1/x
Putting f’(x) = 0
1/x = 0
x = ∞ (this is not defined for x)
So, f(x) does not have minima or maxima
f(x) = x5 – 5x4 + 5x3 – 1. The local maxima of the function f(x) is at x =
f(x) = x5 - 5x4 + 5x3 - 1
On differentiating w.r.t.x, we get
f'(x) = 5x4 - 20x3 + 15x2
For maxima or minima,f'(x) = 0
⇒ 5x4 - 20x3 + 15x2 = 0
⇒ 5x2(x2-4x+3) = 0
⇒ 5x2(x-1)(x-3) = 0
⇒ x = 0,1,3
So, y has maxima at x = 1 and minima at x = 3
At x = 0, y has neither maxima nor minima, which is the point of inflection .
f(1) = 1-5+5-1=0,which is local maximum value at x = 1.
How many units should be sold so that a company can make maximum profit if the profit function for x units is given by p(x) = 25 + 64x - x2
Find two numbers whose sum is 24 and product is a large as possible.
A point c in the domain of a function f is called a critical point of f if
A point C in the domain of a function f at which either f(c) = 0 or f is not differentiable.
The point f is called the critical point.
c is called the point of local maxima
If f ′(x) changes sign from positive to negative as x increases through c, that is, if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c.
c is called the point of local minima
If f ′(x) changes sign from negative to positive as x increases through c, that is, if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c.
c is called the point of inflexion
If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum
two positive numbers x and y are such that x + y = 60.
x + y = 60
⇒ x = 60 – y ...(1)
Let P = xy3
∴ P =(60 – y)y3 = 60y3 – y4
Differentiating both sides with respect to y, we get
For maximum or minimum dP/dy = 0
⇒ 180y2 - 4y3 = 0
⇒ 4y2 (45 - y) = 0
⇒ y = 0 or 45 - y = 0
⇒ y = 0 or y = 45
⇒ y = 45 (∵ y = 0 is not possible)
Thus, the two positive numbers are 15 and 45.