1. how many zeroes does e^sin(x) have?
2. x^3 - 3x^2 - 24x +11 what are the horizontal tangent equations and coordinates of where it touches this graph.
2. x^3 - 3x^2 - 24x +11 what are the horizontal tangent equations and coordinates of where it touches this graph.
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1 e^sin(x) is always > 0 so no zeroes.
2. y = x^3 - 3x^2 - 24x +11 or
dy/dx = 3x^2 -6x -24 = 0 or
x^2 - 2x - 8 = 0 or
x^2 - 4x +2x -8 = 0 or
(x-4)(x+2) = 0 or
x = 2 and x = -4
So at x =2 and x = -4 are points of contact of the horizontal tangents, Corresponding y coordinates are obtained by putting these values in expression for y as
x =2 gives y = 8-12-48+11 = -41 and x = -4 gives y = -64-48+96+11 = -5
So equations are y = -41 which touches at (2, -41) and
y = -5 which touches at (-4, -5)
2. y = x^3 - 3x^2 - 24x +11 or
dy/dx = 3x^2 -6x -24 = 0 or
x^2 - 2x - 8 = 0 or
x^2 - 4x +2x -8 = 0 or
(x-4)(x+2) = 0 or
x = 2 and x = -4
So at x =2 and x = -4 are points of contact of the horizontal tangents, Corresponding y coordinates are obtained by putting these values in expression for y as
x =2 gives y = 8-12-48+11 = -41 and x = -4 gives y = -64-48+96+11 = -5
So equations are y = -41 which touches at (2, -41) and
y = -5 which touches at (-4, -5)
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Sorry for the mistake x = -2 gives y = 39 and x = 4 gives y = -69, y = 39, (-2,39) & y = -69, (4,-69)
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" ninja boy " ...you should know that e^(f(x)) > 0...and you can find the roots of a quadratic to
answer the 2nd part !
{ ( 4 , y(4)) & ( -2 , y(-2)) }
answer the 2nd part !
{ ( 4 , y(4)) & ( -2 , y(-2)) }