Advertisements
Advertisements
प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Advertisements
उत्तर
We have, f (x) = 10 - (x - 1)2 for all x ∈ R
since, (x - 1)2 ≥ 0 ∀ x ∈ R
= - (x - 1)2 ≤ 0 ∀ x ∈ R
= 10 - (x - 1)2 ≤ ∀ x ∈ R
∴ Maximum f (x) = 10 which occurs when x - 1 = 0 i.e, when x = 1
f (x) has no minimum value for, f (x) → - ∞ As |x| →∞
APPEARS IN
संबंधित प्रश्न
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 maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
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:
f(x) = x3 − 6x2 + 9x + 15
Prove that the following function do not have maxima or minima:
f(x) = ex
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
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.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Divide the number 20 into two parts such that sum of their squares is minimum.
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?
If f(x) = x.log.x then its maximum value is ______.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The function y = 1 + sin x is maximum, when x = ______
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
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 A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
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 greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
If \[f\] has a maximum value at \[c\], what is \[c\] called?
What is an absolute minimum?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from negative to positive as \[x\] passes through \[c\], what is \[c\]?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
What are the critical points obtained from \[12x(x-1)(x+2)=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f''(x)\]?
