f(x)=(x-a)(2x-a)...(nx-a)\nderive
121240
f'(x) = 1*(2x-a)(3x-a) ...(nx-a) + (x-a)*2*(3x-a)...(nx-a) +(x-a)(2x-a)*3*(4x-a)...(nx-a)+ ... + (x-a)(2x-a)...((n-1)x-a)*n
565431
2x<1 , x<0.5\n -1<x\n\n     -1< x<0.5
446646
f(x) = -3x^2 - 6x - 5\n\ny-intercept (0, -5)\nno x-intercept\n\nvertex (-1, -2)\n\naxis x = -1\n\ndomain (-∞, ∞)\n\nrange (-∞, -2]
477040
f(x)=x^3+5x^2+3x+1\nf'(x)=3x^2+10x+3\nsetting it to zero\n(3x+1)(x+3)=0\nx=-1/3 and x=-3\nf''(x)=6x+10\nat x=-1/3 f''(x)=+ve and so min\nat x=-3 f''(x)=-ve and so max
539771
f(x) = x(x+2)(x-2)(x-4)\n\nMultiply it out if you need it in general form.\n\nThere could also be a constant in front of the x.
501864
Your correct, it is equal to f(x).\n\nthe integral and the differential cancel eachother out.!
174019
1) [1/x] <= 1/x, for all x>0\n2) [1/x] = 1/x, iff 1/x is an integer, so x=1/n for some integer n.\n3) [1/x] < 1/x, for all x != 1/n for any n (x>0).\n4) So, f(x) = [x*[1/x]] = [x*1/x] = [1] = 1 for all x=1/n for some integer n.\n5) Also, x*[1/x] < x*1/x = 1 for all x != 1/n for any n (x>0),\nthen f(x) = [x*[1/x]] = [<1] = 0 for all x != 1/n for any n (x>0).\n\nlim f(x)(x->a) = b, iff for ALL sequences {a_n->a} f(a_n)->b.\n\nLet's take 2 sequences:\n{a_n = 1/n} f(a_n)=1 for all n (see 4), so f(a_n)->1.\n{b_n = 2/(2*n+1)} f(b_n)=0 for all n (see 5), so f(b_n)->0.\nEasy to see that {a_n}->0, {b_n}->0.\n\nThus, there is NO limit of f(x) when x->0.\n\nPS: "iff" means "if and only if"
518349
Well, let's look at a few of them to see if there is a pattern.\nf(x) = 2x + 1 = (2^1)x + (2^1 -1)\n(fof)(x) = 2(2x+1) + 1 = 4x + 3 = (2^2)x + (2^2 -1)\n(fofof)(x) = 4(2x+1) + 3 = 8x + 7 = (2^3)x + (2^3 -1)\n(fofofof)(x) = 8(2x+1) + 7 = 16x + 15 = (2^4)x + (2^4 -1)\nDo you see the pattern now?\n(fofofo...of)(x) (n functions) = (2^n)x + (2^n -1)
596341
given, F(x)= x-[x]\n=> F(1/x)= 1/x- [1/x]\n\nthen equation to be solved is :\n\nx-[x]+ 1/x -  [1/x]=1\n\nor x + 1/x = 1 + [x] + [1/x]\n\nnow when x>0, [x] = x \nthen from above equation,\nx+ 1/x= x+1/x+1  this is not possible for any positive x.\n\nwhen x<0,  [x]= -x\n\nthen,\nx+ 1/x = -x -1/x +1\nor, x+ 1/x = 1/2, this has no solution such that x is negative.\n\nHENCE the above equation has no solution.
523560
Ah ha, now we see the real question.\n\nThis is an application of the product rule.\n\ng(x) = xf(x)\ng'(x) = xf'(x) + (1)f(x)  [by the product rule]\ng'(2) = 2f'(2) + f(2) = 2(-5) + 3 = -7\n\nAlso,\n\ng(2) = 2f(2) = 2(3) = 6\n\nSo the point is (2, 6), and the slope is -7....\n\ny - 6 = -7(x - 2)
387403
f(x) = (x+3)(x^2 + 16)
342377
wx - x =\n\nx(w - 1)\n\n- - - - - - -s-
412457
1) If h(x) = f(g(x)) = (x^3 -1)/(x^3 +1), then \nf(x) = (x-1)/(x+1) and g(x) = x^3\n2) If h(x) = f(g(x)) = 3sqrt(x^2 -8), then\nf(x) = 3sqrt(x) and g(x) = x^2 -8
343741
a function is written in  the form f(x) or g(X)  or anything relating to that form\nan equation has and equal sign\n\ntherefore this one is a linear equation
125669
x=A*A\n\nSO\n\nA*A + ? = (A+1)*(A+1) = A*A+2A+1\n\nSO ? = 2A+1\n\n\nand sice you defined X=A*A , A=√(X)\n\nSO ? = 2A + 1 = 2√(X) + 1
457242
(3x - a)(x - 2)(x - 7) = \n(3x - a)(x^2 - 7x - 2x + 14) = \n(3x - a)(x^2 - 9x + 14) = \n3x^3 - 27x^2 + 42x - ax^2 + 18ax - 14a\n3x^3 + (-27 - a)x^2 + (42 + 18a)x - 14a\n\n3x^3 - 32x^2 + 81\n\nso\n\n(-27 - a)x^2 + (42 + 18a)x - 14a = -32x^2 + 81\n(-27 + 32 - a)x^2 + (42 + 18a)x + (-14a - 81) = 0\n(5 - a)x^2 + (42 + 18a)x - (14a + 81) = 0\n(-a + 5)x^2 + (18a + 42)x - (14a + 81) = 0\n-(a - 5)x^2 + (18a + 42)x - (14a + 81) = 0\n\nx = (-b ± sqrt(b^2 - 4ac))/(2a)\n\nx = (-(18a + 42) ± sqrt((18a + 42)^2 - 4(-(a - 5)(-(14a + 81)))/(2(-(a - 5))\n\nx = (-(18a + 42) ± sqrt(((18a + 42)(18a + 42)) - 4(a - 5)(14a + 81)))/(-2(a - 5))\n\nx = (-(18a + 42) ± sqrt((324a^2 + 756a + 756a + 1764) - 4(14a^2 + 81a - 70a - 405))) / (-2(a - 5))\n\nx = (-(18a + 42) ± sqrt(324a^2 + 1512a + 1764 - 4(14a^2 + 9a - 405))) / (-2(a - 5))\n\nx = (-(18a + 42) ± sqrt(324a^2 + 1512a + 1764 - 56a^2 - 36a + 1620))) / (-2(a - 5))\n\nx = (-(18a + 42) ± sqrt(268a^2 + 1476a + 3384)) / (-2(a - 5))\n\nx = (-6(3a + 14) ± sqrt(4(67a^2 + 369a + 846)))/(-2(a - 5))\n\nx = (-6(3a + 14) ± 2sqrt(67a^2 + 369a + 846)) / (-2(a - 5))\n\nx = (3(3a + 14) ± sqrt(67a^2 + 369a + 846))/(a - 5)\n\nUnless you mistyped something, 3x^3 - 32x^2 + 81 does not factor into (3x - a)(x - 2)(x - 7)\n\n--------------------------------------------------------------------------------\n\nx^3 - x^2 + 16x - 16\n(x^3 - x^2) + (16x - 16)\nx^2(x - 1) + 16(x - 1)\n(x^2 + 16)(x - 1)\n\nif you were to take this further, then\n\n(x - 4i)(x + 4i)(x - 1)
656547
=(x+3)(x-1) - (x-2)(x+4)  /  (x+4)(x-1)\n=(x^2 +2x - 3) - (x^2 + 2x - 8) / (x^2 + 3x - 4)\n=(5) / (x^2 + 3x - 4)