Question

Find min and max value of 12 cosx+5sinx +4

Answer 1

Answer:

Max

Step-by-step explanation:

Answer 2

Answer:

maximum value = 17

minimum value = -9

Step-by-step explanation:

Here

$f(x) = 12cos(x) +5sin(x)+4$

Now

$\frac{d}{dx}f(x) = 12*(-sin(x))+5cos(x)\\\frac{d}{dx}f(x) = -12sin(x) + 5cos(x)$

Now equating the above first derivative to 0

$\frac{d}{dx}f(x) = 0\\-12sin(x) + 5cos(x) = 0\\5cos(x) = 12sin(x)\\\frac{5}{12} = \frac{sin(x)}{cos(x)}\\tan(x) = \frac{5}{12}\\x = tan^{-1}(\frac{5}{12}) = 22.619$

Now calculating second derivative of f(x)

$\frac{d}{dx}f(x) = -12sin(x) + 5cos(x)\\\frac{d}{dx}\frac{d}{dx}f(x) = -12cos(x) + 5*(-sin(x))\\\\\frac{d^{2}}{dx^{2}}f(x) = -12cos(x)-5sin(x)$

Now finding value of second derivative at x = 22.619

$\frac{d^2}{dx^2}f(22.619) = -12cos(22.619) - 5sin(22.619)\\\frac{d^2}{dx^2}f(22.619) = - 11.07 - 1.92 = -12.99$

Since the value coming out is negative therefore f(22.619) is the maxima

lets calculate maxima

$f(22.619) = 12cos(22.619) + 5sin(22.619)+4\\= 11.07 + 1.92 + 4 = 16.99\\= 17 (approx.)$

therefore the function has maximum value of 17

since the function is like a wave in the graphical form

it will oscillate between maximum and minimum value after every 180 degrees since we found it maximum value at x = 22.619

therefore it will show it minimum value at 22.619±180

so let us calculate minimum value by keeping x = 22.619+180 = 202.619

therefore

$f(202.619) = 12cos(202.619)+5sin(202.619)+4\\= - 11.07 -1.92 + 4 = - 8.99 = -9 (approx.)$

therefore minimum value is -9

also you can have a better visual using a graphing calculator