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Given the function f( x ) = 2.3 \sin ( x ) + 3.4 \cos ( x ) . Find the maximum and minimum...

Question:

Given the function {eq}f( x ) = 2.3 \sin ( x ) + 3.4 \cos ( x ) {/eq}. Find the maximum and minimum value of this function?

Finding Maximum Value

With the first derivative of a function with one independent variable, we can find its maximum and minimum values, because the first derivative is the tangent line's slope.

Answer and Explanation:

The function has maximum and minimum values at the first derivative zero.

{eq}\displaystyle \ f'(x) \; = \; 2.3\,\cos \left( x \right) - 3.4\,\sin \left( x \right) \\ {/eq}

Then {eq}\displaystyle 0= 2.3\,\cos \left( x \right) - 3.4\,\sin \left( x \right) \; \Leftrightarrow \; x=0.5947592575 \; \text{ & } \; x= 3.736351911 \\ \; \text{ or } \; x=0.5947592575 +\pi \; \text{ & } \; x= 3.736351911+\pi \\ \; \text{ or } \; x=0.5947592575 +2\pi \; \text{ & } \; x= 3.736351911+2\pi \\ \; \text{ or } \; x=0.5947592575 +\pi \mathbb{N} \; \text{ & } \; x= 3.736351911+\pi \mathbb{N} \\ {/eq}

Remember, trigonometric functions has repetitive behavior. Then, if we choose a closed interval we can identify maximum and minimum values; we choose {eq}\displaystyle x \; \in [0,2\pi] {/eq}

Increasing and decreasing intervals:

{eq}\begin{array}{c} \; \text{Interval} \; & { 0 < x < 0.594759258 } & { 0.594759258 < x < 3.736351911 } & { 3.736351911 < x < 2\pi } \\ \hline Test \space{} value & { x = 0.5 } & { x = 2 } & { x = 6 } \\ Test & { f'( 0.5 ) = + > 0 } & { f'( 2 ) = - < 0 } & { f'( 6 ) = + > 0 } \\ Conclusion & { increasing } & { decreasing } & { increasing } \\ \end{array} \\ \therefore \boxed { \text{ decreasing's interval(s)} \; \ ( 0.594759258 , 3.736351911 )} \\ \therefore \boxed { \text{ increasing's interval(s)} \; \ ( 0 , 0.594759258) \cup ( 3.736351911 , 2\pi )} \\ {/eq}

Therefore, the function has maximum value at {eq}x=0.594759258 {/eq} and it is {eq}\displaystyle f(0.594759258)= 4.104875151 \\ {/eq}

and, the function has minimum value at

{eq}x=3.736351911 {/eq}

And, it is

{eq}\displaystyle f(3.736351911)=- 4.104875150 \\ {/eq}


The function and its maximum and minimum values



Learn more about this topic:

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Find the Maximum Value of a Function: Practice & Overview

from CAHSEE Math Exam: Tutoring Solution

Chapter 10 / Lesson 12
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