Advertisements
Advertisements
प्रश्न
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
उत्तर
\[\text { Given }: \hspace{0.167em} M = \frac{WL}{2}x - \frac{W}{2} x^2 \]
\[ \Rightarrow \frac{dM}{dx} = \frac{WL}{2} - 2 \times \frac{Wx}{2}\]
\[ \Rightarrow \frac{dM}{dx} = \frac{WL}{2} - Wx\]
\[\text { For maximum or minimum values of M, we must have }\]
\[\frac{dM}{dx} = 0\]
\[ \Rightarrow \frac{WL}{2} - Wx = 0\]
\[ \Rightarrow \frac{WL}{2} = Wx\]
\[ \Rightarrow x = \frac{L}{2}\]
\[\text { Now,} \]
\[\frac{d^2 M}{d x^2} = - W < 0\]
\[\text { So,M is maximum at }x = \frac{L}{2} . \]
APPEARS IN
संबंधित प्रश्न
f(x)=| x+2 | on R .
f(x) = (x \[-\] 5)4.
f(x) = x3 (x \[-\] 1)2 .
f(x) = \[\frac{1}{x^2 + 2}\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
`f(x)=xsqrt(1-x), x<=1` .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
If x+y=8, then the maximum value of xy is ____________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
