Advertisements
Advertisements
Question
Prove that the following function do not have maxima or minima:
f(x) = ex
Advertisements
Solution
Given function, f‘(x) = ex
∴ f‘(x) = ex
= f' (x) = ex ∀ x ∈ R
f' (x) = ex > 0 ∀ x ∈ R
f has no critical point.
Thus, there is no point at which f can have an extremum.
∴ f has neither a maximum nor a minimum.
APPEARS IN
RELATED QUESTIONS
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
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`
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) = x/2 + 2/x, x > 0`
Prove that the following function do not have maxima or minima:
g(x) = logx
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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]`
What is the maximum value of the function sin x + cos x?
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3.`
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
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.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
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 and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
The maximum value of `(1/x)^x` is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
The minimum value of 2sinx + 2cosx is ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
If x + y = 8, then the maximum value of x2y is ______.
Divide the number 100 into two parts so that the sum of their squares is minimum.
