For what value of a is the line that is tangent to the function when x=pi/6 see below for information

Question

a) For what value of a is the line -x+2y= -3/2√3 -pi/6 tangent to the function y=acos² x when x= pi/6 use exact values only

b) verify that with your value of a, the tangent line is in fact tangent to the curve at x=pi/6

Answer 1

a)

-x+2y=-3/(2√3)-π/6

f(x) = y = x/2-3/(4√3)-π/12

g(x) = y = acos²(x)

when x = π/6

Find when f(π/6)=g(π/6)

π/12-3/(4√3)-π/12=a(√3/2)²

a=-1/√3

b)

f '(x) = 1/2

g'(x) = (-1/√3)(2cos(x))(-sin(x))

g'(π/6) = (1/√3)(√3)(1/2) = 1/2 = f'(π/6)

Therefore g(x) is tangent to f(x) at π/6 since the tangent of those two functions are equal at that point.

Answer 2

a) y' = - 2acosxsinx = - asin2x; y'(π/6) = - asinπ/3 = - a√3/2 = 1/2; a = - √3/3;

b) x = π/6; y = - √3/3cos²x; y = - √3/3cos²π/6 = - √3/4; y' = √3/3sin2x; y' = √3/3sinπ/3 = 1/2

Equation of tangent line at x = π/6: y = - √3/4 + 1/2(x - π/6); 2y = - √3/2 + x - π/6; - x + 2y = - √3/2 - π/6