+537 Find the minimum value of \(4 \cos \theta + \frac{49}{\cos \theta}\)
For \(0 \le \theta < \pi/2\)
Thanks!
By the AM-GM inequality, \(4 \cos \theta + \frac{49}{\cos \theta} \ge 2 \sqrt{4 \cos \theta \cdot \frac{49}{\cos \theta}} = 28\)
The minimum value is 28.
+129899 y = 4cosθ + 49 / cosθ
y = 4cosθ + 49secθ take the derivative
y' = -4sinθ + 49secθtanθ
y'= -4sinθ + 49sinθ /cos^2θ set this to 0
-4sinθ + 49sinθ /cos^θ = 0
49sinθ / cos^2θ = 4sinθ
49sinθ = 4sinθcos^2θ
49sinθ - 4sinθcos^2 = 0
sinθ (49 - 4cos^2θ) = 0
sinθ (7 - 2cosθ)(7 + 2cosθ) = 0
So either
sinθ = 0 7 - 2cosθ = 0 7 + 2cosθ = 0
θ = 0 7 = 2cosθ 2cosθ = -7
cosθ = 7/2 ( impossible) cosθ = -7/2 (impossible)
So....the minimum occurs when θ = 0....
And the minimum = 4cos(0) + 49/cos(0) = 4 + 49/1 = 53
As this graph shows : https://www.desmos.com/calculator/42hzhstsx6
