00:01
Hi everyone, welcome to the problem.
00:02
So in this problem, they have given us y is equal to f double dash of x.
00:07
In the graphs, they have represented y is equal to f double dash of x.
00:12
And from that, we have to find the intervals of concavity for the function f of x and then find the points of inflection for f of x.
00:20
So coming to the first a part.
00:25
So from the graph, we find that f double dash of x is nothing but one.
00:31
Okay, so in order to find f dash of x from f double dash of x, we have to integrate f double dash of x with respect to dx.
00:42
So f double dash of x is 1.
00:45
So 1 multiplied by d x.
00:50
So you will be getting f dash of x as x plus c on.
00:56
Now in order to get f of x, we have to integrate f dash of x with respect to d x.
01:03
So integration of x plus c1.
01:07
So f of x will be x to the part 2 plus c1 multiplied by x plus c2.
01:17
So if you see from here f of x is a parabola, it is a parabola.
01:25
And since an f double dash of x, it's greater than zero.
01:30
Okay, so it means that f of x is a upward.
01:36
Facing is an upward facing parabola so from this we can say that f of x is conch upwards so the answer to this problem is f of x is concave upwards and the point of inflections the point of inflection for f of x occur in the parabola so this is the answer for part a.
02:35
So moving on to part b.
02:40
From the graph, we can find that f double dash of x is equal to x minus 2.
02:46
So in order to find f dash of x from f double dash of x, we have to integrate f double dash of x with respect to dx.
02:55
So integration of x minus 2 multiplied by dx.
03:00
So if dash of x will be x, the part of the part of, 2 divided by 2 minus 2 multiplied by x plus c1.
03:11
Now to get f of x from f dash of x we have to integrate f dash of x with respect to d x.
03:19
So we will be getting integration of x to the part 2 divided by 2 minus 2x plus c1 with respect to d x.
03:29
So our f of x will be x to the part 3 divided by 6 minus 2 to the 2 multiplied by x to the part 2 plus c1 x plus c2 so our f of x will be 2 2 2 will cancel x to the part 3 divided by 6 minus x to the part 2 plus c1 x plus c2 so if you see here, f of x is a cubic polynomial.
04:05
So f of x is a cubic polynomial.
04:10
And if double dash of x is lesser than zero, for all x belongs to minus infinity comma 2.
04:22
So f double dash of x changes its sign when x is equal to 2.
04:33
So therefore, from this, we can say that the answer, this is probably b part is, f of x is concave downwards and the point of inflection for f of x occur when x is equal to 2.
05:08
So this is the answer for the b part.
05:11
Next, moving on to the c part.
05:15
From the graph we can find that f double dash of x is equal to x minus 2 multiplied by x plus 1 so that is equal to x squared plus x minus 2 x minus 2 which is equal to x square minus x minus 2 so in order to find f dash of x we have to integrate f double dash of x with respect to d x so that is equal to integration of x squared minus x minus 2 d x that is equal to x to the part 3 divided by 3 minus x to the part 2 minus 2 plus c 1 now in order to find f of x we have to integrate f dash of x with respect to d x so that is equal to integration of x to the part 3 divided by 3 minus x to the part 2 divided by 2 minus 2 x to the plus c1 multiplied by d x okay f of x is equal to equal to x to the power four divided by 12 minus x to the part 3 divided by 6 minus 2 x to the part 2 divided by 2 plus c1 x plus c2 so 2 2 gets cancelled so that is equal to f of x is equal to x to the power 4 divided by 12 minus x to the part 3 divided by 6 minus x to the part 2 plus c1x plus c2 this is our f of x so from this we can say that if double dash of x is lesser than 0 for all x belongs to minus infinity comma minus 1 so from this f double dash of x changes sign at x is equal to minus 1...