Advertisements
Advertisements
Question
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Advertisements
Solution
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point `(- 1/3, (-74)/9)`.
Explanation:
We have y = 4x2 + 2x – 8 .....(i)
And y = x3 – x + 13 .....(ii)
Differentiating eq. (i) w.r.t. x, we have
`"dy"/'dx"` = 8x + 2
⇒ m1 = 8x + 2 .....[m is the slope of curve (i)]
Differentiating eq. (ii) w.r.t. x, we get
`"dy"/"dx"` = 3x2 – 1
⇒ m2 = 3x2 – 1 ......[m2 is the slope of curve (ii)]
If the two curves touch each other, then m1 = m2
∴ 8x + 2 = 3x2 – 1
⇒ 3x2 – 8x – 3 = 0
⇒ 3x2 – 9x + x – 3 = 0
⇒ 3x(x – 3) + 1(x – 3) = 0
⇒ (x – 3)(3x + 1) = 0
∴ x = 3, `(-1)/3`
Putting x = 3 in equation (i), we get
y = 4(3)2 + 2(3) – 8
= 36 + 6 – 8
= 34
So, the required point is (3, 34)
Now for x = `- 1/3`
y = `4((-1)/3)^2 + 2((-1)/3) - 8`
= `4 xx 1/9 - 2/3 - 8`
= `4/9 - 2/3 - 8`
= `(4 - 6 - 72)/9`
= `(-74)/9`
∴ Other required point is `(- 1/3, (-74)/9)`.
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).
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 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:
g(x) = x3 − 3x
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
Prove that the following function do not have maxima or minima:
f(x) = ex
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
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) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = x log x
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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 the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
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.
Determine the maximum and minimum value of the following function.
f(x) = x log x
Divide the number 20 into two parts such that their product is maximum.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area 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`
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 ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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.
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
A function f(x) is maximum at x = a when f'(a) > 0.
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 ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` 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)
