3 calculus curve questions?

1) f(x) = 2x^3 - 3x² - 12x

f'(x) = 6x² - 6x - 12

f''(x) = 12x - 6

2) f(x) = sinx + cosx

f'(x) = (sinx)' + (cosx)'

= cosx - sinx

f''(x) = (cosx)' - (sinx)'

= - sinx - cosx

3) f(x) = (2x² - 5x +2)/(x+2)

f'(x) = [(2x² - 5x + 2)'(x+2) - (2x² - 5x + 2)(x + 2)' ] / (x + 2)²

= [ ( 4x - 5 )(x + 2) - 2x² + 5x - 2 ] / (x + 2)²

= ( 4x² + 8x - 5x - 10 - 2x² + 5x - 2 ) / (x + 2)²

= ( 2x² + 8x - 12 ) / (x + 2)²

f''(x) = { (2x² + 8x - 12)'(x + 2)² - (2x² + 8x - 12)[(x + 2)²]' } / (x + 2)^4

= [ (4x + 8)(x + 2)² -2(2x² + 8x - 12)(x + 2) ] / (x + 2)^4

= (x + 2) [ (4x + 8)(x + 2) - 2(2x² + 8x - 12) ] / (x + 2)^4

= [ (4x + 8)(x + 2) - 2(2x² + 8x - 12) ] / (x + 2)^3

= ( 4x² + 8x + 8x + 16 - 4x² - 16x + 24 ) / (x + 2)^3

= 40 / (x + 2)^3

I probably do not provide complete solutions simplest aid to discover solutions your self. However, this time I ought to step in to proper mistakes above. B. dy/dx = 4x^three + 12x^two = 4x^two(x + three) dy/dx = zero way x = zero or x = -three. So there are turning elements at (zero,zero) and (-three, -27) d2y/dx2 = 12x^two + 24x. At x = -three d2y/dx2 = 36 so optimistic way a minimal. nonetheless, at x = zero, d2y/dx2 = zero and whilst dy/dx and d2y/dx2 are each zero it will imply that this can be a minimal or highest or factor of inflection. You ought to evaluate the gradient both aspect or y importance both aspect. x = one million offers y = five, x = zero offers y = zero and x = -one million offers y = -three so this can be a factor of inflection. C. d2y/dx2 = 12x^two + 24x = 12x(x + two). d2y/dx2 = zero whilst x = zero or x = -two. The factor (-two, -sixteen) is undoubtedly a factor of inflection however observe remark above approximately whilst x = zero. As proven above it seems to be an additional factor of inflection. The major factor is that d2y/dx2 = zero does no longer consistently imply you've got a factor of inflection. The role y = x^four is a sensible evidence of this. It has d2y/dx2 = zero on the foundation however this can be a minimal.

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