The greatest value of 5 sin2 x + 7 cos2 x - 4 sin x cos x will be:1. 6

1.	The greatest value of 5 sin2 x + 7 cos2 x - 4 sin x cos x will be:1. 6 - √52. 6 + √53. -6 + √54. -6 - √5
Answer» Correct Answer - Option 2 : 6 + √5 Concept: Following steps to finding maxima and minima using derivatives. 1. Find the derivative of the function. 2. Set the derivative equal to 0 and solve. This gives the values of the maximum and minimum points. 3. Now we have to find the second derivative. I. f``(x) is less than 0 then the given function is said to be maxima II. If f``(x) Is greater than 0 then the function is said to be minima sin x = cos x when x = π/4 Short trick: The maximum value of the trigonometric function \({\bf{a}}\;{\bf{sin}}\;{\bf{x}}\; + \;{\bf{b}}\;{\bf{cos}}\;{\bf{x}}\) is \(\sqrt {{{\bf{a}}^2} + {{\bf{b}}^2}} \) Calculation: Given: y = 5 sin² x + 7 cos² x – 4 sin x cos x y = (2 sin x – cos x)² + sin² x + 6 cos² x y = (2 sin x – cos x)² + (sin² x + cos² x) + 5 cos² x \(y_{max}=[\sqrt{(2)^2+(1)^2 )}]+1+5\) =6+√5

The greatest value of 5 sin2 x + 7 cos2 x - 4 sin x cos x will be:1. 6 - √52. 6 + √53. -6 + √54. -6 - √5

Answer» Correct Answer - Option 2 : 6 + √5

Concept:

Following steps to finding maxima and minima using derivatives.

1. Find the derivative of the function.

2. Set the derivative equal to 0 and solve. This gives the values of the maximum and minimum points.

3. Now we have to find the second derivative.

I. f``(x) is less than 0 then the given function is said to be maxima

II. If f``(x) Is greater than 0 then the function is said to be minima

sin x = cos x when x = π/4

Short trick:

The maximum value of the trigonometric function

\({\bf{a}}\;{\bf{sin}}\;{\bf{x}}\; + \;{\bf{b}}\;{\bf{cos}}\;{\bf{x}}\) is \(\sqrt {{{\bf{a}}^2} + {{\bf{b}}^2}} \)

Calculation:

Given:

y = 5 sin² x + 7 cos² x – 4 sin x cos x

y = (2 sin x – cos x)² + sin² x + 6 cos² x

y = (2 sin x – cos x)² + (sin² x + cos² x) + 5 cos² x

\(y_{max}=[\sqrt{(2)^2+(1)^2 )}]+1+5\)

=6+√5