Advertisements
Advertisements
Question
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Advertisements
Solution
f(x) = x3 - 9x2 + 24x
∴ f '(x) = 3x2 - 18x + 24
∴ f ''(x) = 6x - 18
Consider, f '(x) = 0
∴ 3x2 - 18x + 24 = 0
∴ 3(x2 - 6x + 8) = 0
∴ 3(x - 4)(x - 2) = 0
∴ (x - 4)(x - 2) = 0
∴ x = 2 or x = 4
For x = 4,
f ''(4) = 6(4) - 18 = 24 - 18 = 6 > 0
∴ f(x) is minimum at x = 4
∴ Minima = f(4) = (4)3 - 9(4)2 + 24(4)
= 64 - 144 + 96 = 16
For x = 2,
f ''(2) = 6(2) - 18 = 12 - 18 = - 6 < 0
∴ f(x) is maximum at x = 2
∴ Maxima = f(2) = (2)3 - 9(2)2 + 24(2) = 8 - 36 + 48 = 20
∴ Maxima = 20 and Minima = 16
APPEARS IN
RELATED QUESTIONS
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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) = x3 − 3x
Prove that the following function do not have maxima or minima:
g(x) = logx
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
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.
Show that among rectangles of given area, the square has least perimeter.
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)`.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
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 function y = 1 + sin x is maximum, when x = ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Range of projectile will be maximum when angle of projectile is
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 ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
The minimum value of 2sinx + 2cosx is ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
