Advertisements
Advertisements
प्रश्न
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`h(x) = sinx + cosx, 0 < x < pi/2`
Advertisements
उत्तर
given function, `h(x) = sin x + cos x, 0 < x < pi/2`
= h' (x) = cos x - sin x for all `x in (0, pi/2)`
For critical points, let h'(x) = 0
= cos x - sinx = 0
∴ h'(x) = cos x - sin x = cos x (1 - tan x)
`= tan x = 1 = x pi/4`
At x = `pi/4`, if the value of x is kept a little less than `pi/4`, then tan x will be less than 1 and if the value of x is kept a little more than `pi/4`, then tan x will be more than 1.
Thus, the sign of 1 - tan x changes from positive to negative and there is no change in sign in cos x.
Hence, x = `pi/4,`h is maximum.
Local maximum value = h `= (pi/4) = sin pi/4 + cos pi/4`
`= 1/sqrt2 + 1/sqrt2`
`= 2/sqrt2`
`= sqrt2`
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Find the maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = 1/(x^2 + 2)`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
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.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
Divide the number 20 into two parts such that their product is maximum
A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.
Solution: Let the dimensions of the rectangle be x cm and y cm.
∴ 2x + 2y = 36
Let f(x) be the area of rectangle in terms of x, then
f(x) = `square`
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme value, f'(x) = 0, we get
x = `square`
∴ f''`(square)` = – 2 < 0
∴ Area is maximum when x = `square`, y = `square`
∴ Dimensions of rectangle are `square`
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) `= x sqrt(1 - x), 0 < x < 1`
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
