Question

Find the maximum and minimum value of the function f(x) = 5 sinx + 12 cosx + 10​

Answer 1

Answer:

The maximum value of the function f(x) = 5 sinx + 12 cosx + 10​ is 22.29

explanation:

• The maximum and minimum point of a function is the points where the function neither increase nor decrease

• For a maximum point of a function Y, the second derivative of the function is greater than zero, ($\frac{d^{2}Y }{dx^{2} }$  is positive and for minimum point, it is less than zero($\frac{d^{2}Y }{dx^{2} }$ is negative)

STEP 1

The given function is $f(x)=5sinx+12cosx+10$

the first derivative is $\frac{df(x)}{dx} =5cosx-12 sinx$

STEP 2

equate the first derivative to zero to get the value of x

⇒

$5cosx-12 sinx=0\\5cosx=12 sinx$

⇒

$sinx/cosx=5/12\\tanx=0.41\\x=22.29$

STEP 3

put the value of x we get

$f(22.2)=5sin22.2+12cos22.2+10=22.8$

which is the maximum value of the function

Answer 2

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: f(x) = 5sinx + 12cosx + 10$

$\rm \: Divide \: and \: multiply \: by \: \sqrt{ {5}^{2} + {12}^{2} } = \sqrt{25 + 144} = \sqrt{169} = 13$

So, above expression can be rewritten as

$\rm \: f(x) = 13\bigg(\dfrac{5}{13}sinx + \dfrac{12}{13}cosx \bigg) + 10$

Let assume that

$\rm \: \dfrac{5}{13} = cosy$

So,

$\rm \: siny = \sqrt{1 - {cos}^{2}y } = \sqrt{1 - \dfrac{25}{169} } = \sqrt{ \dfrac{144}{169} } = \dfrac{12}{13}$

So, above expression can be rewritten as

$\rm \: f(x) = 13(sinx \: cosy \: + \: siny \: cosx) + 10$

$\rm \: f(x) = 13sin(x + y) + 10 \: \: where \: y \: = \: {tan}^{ - 1} \dfrac{5}{12}$

Now, we know that

$\rm \: - 1 \leqslant sin(x + y) \leqslant 1$

So,

$\rm \: - 13 \leqslant 13 \: sin(x + y) \leqslant 13$

On adding 10 in each term, we get

$\rm \: - 13 + 10 \leqslant 13 \: sin(x + y) + 10\leqslant 13 + 10$

$\rm\implies \: - 3 \leqslant f(x) \leqslant 23$

So,

$\rm\implies \:Maximum \: value \: of \: f(x) = 23$

and

$\rm\implies \:Minimum \: value \: of \: f(x) \: = \: - \: 3$

$\rule{190pt}{2pt}$

Short Cut Trick

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf Function & \bf Range \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf asinx + bcosx & \sf [ - \sqrt{ {a}^{2} + {b}^{2} }, \: \: \sqrt{ {a}^{2} + {b}^{2} }] \\ \\ \sf asinx + bcosx + c & \sf [c - \sqrt{ {a}^{2} + {b}^{2} }, \: c + \sqrt{ {a}^{2} + {b}^{2} }] \end{array}} \\ \end{gathered} \\ \end{gathered}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Function & \bf Range \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf y = sinx & \sf   - 1 \leqslant y \leqslant 1\\ \\ \sf y = cosx & \sf  - 1 \leqslant y \leqslant 1 \\ \\ \sf y = tanx & \sf y \:  \in \: ( -  \infty , \infty )\\ \\ \sf y = cosec & \sf y \leqslant  - 1 \:  \: or \:  \: y \geqslant 1\\ \\ \sf y = secx & \sf y \leqslant  - 1 \:  \: or \:  \: y \geqslant 1\\ \\ \sf y = cotx & \sf y \:  \in \: ( -  \infty , \infty ) \end{array}} \\ \end{gathered}$