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junior college 2 | H2 Maths
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Need some help in how the equations needed to be derived to the final answer. I have outline my doubts in red. Any help or advice is much appreciated. Thank you in advance.
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eqn of (all the) tangents to the curve C is dy/dx.
(see green box)
and the eqn of the tangent to curve AND // to x-axis is:
y=2x (blue box)
by substituting y=2x into C, we are solving simultaneously the two eqns (see blue and red boxes)
*do u know: in solving 2 simultaneous eqns, we are merely trying to find if there is an intersection point that is common to both eqns?
=> if there is no solution to 2 eqns, this means there is no intersection point between the 2 eqns.
since the solution is: sin(2x) = -x² - 3, we hv to figure out if this is a valid solution.
1. x² is always +ve (plus zero) for all x
2. -x² is always -ve for all x
hence, we can conclude: -x² - 3 is less than or equal to -3. (see yellow box)
therefore, we see the solution here means:
sin(2x) is less than or equal to -3
and do u know: all sinusoidal functions of sin x, cos x, (1)sinx, (1)sinθ, (1)sin5θ, ... etc. is always between the values of -1 and 1?
this applies the same to sin(2x). (see purple box)
thus, we can see that the solution:
" sin(2x) is less than or equal to -3 "
is an invalid ans.
So we conclude :
there is NO tangent to C that is also // to x-axis.
hope this helps.