Question

Functions f(x) and g(x) are shown below:

f(x) = 3x2 + 12x + 16 g(x) = 2 sin(2x - π) + 4

Using complete sentences, explain how to find the minimum value for each function and determine which function has the smallest minimum y-value.

Answer 1

\[\LARGE f(x)= 3x^2 + 12x + 16 \]
To find the minimum value of f(x),We use calculus and differentiate it n equate to 0.
Let it be equal to "y".\[\Large \frac{dy}{dx}=6x+12=0 =>x=-2\]

Answer 2

Similarly for g(x)\[\LARGE g(x) = 2 \sin(2x - π) + 4\]
\[\Large \frac{dy}{dx}=2 \cos(2x-\pi) \times 2 = 0 =>2x-\pi = \frac{\pi}{2}\]
\[\LARGE x = \frac{3 \pi}{4}\]

Answer 3

Now you need to put both the values of x in both the equations respectively to find the minimum value and compare which is lesser as asked in the question.

Answer 4

You can find the 2nd derivative of both the functions to insure that \[\Huge \frac{d^2y}{dx^2}>0\] hence minimima

Answer 5

the function f(x) is an equation of parabola and it has its minimum value for x=-2 and f(x)=4 while the other function g(x) has its minimum value of 2 ... as sine has a smallest value of -1

Answer 6

What are the values of x

Answer 7

-2 and 3pi/4

Answer 8

How did you get them?

Answer 9

Sorry I just don't understand these type of problems