Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and 4x + 1 Select the correct choice below and, if necessary, fill in the answer box within your choice. O A. The function is increasing on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) O B. The function is not increasing on any interval. Select the correct choice below and, if necessary, fill in the answer box within your choice. O A. The function is decreasing on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) O B. The function is not decreasing on any interval. Select the correct choice below and, if necessary, fill in the answer box within your choice. O A. The function is concave up on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) O B. The function is not concave up on any interval. Select the correct choice below and, if necessary, fill in the answer box within your choice. OA. The function is concave down on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) B. The function is not concave down on any interval. Click to select and enter your answer(S).
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
The given function is :
Now,
To determine where the function is
increasing ;
decreasing;
concave up;
concave down;
So,
We use the first derivative test:
Take the derivative of the given expression:
So,
Equating this derivative equal to 0 to get the intervals, we get :
Therefore:
Their will be no interval for the function where the function is increasing or decreasing!
And , hence no critical points exist for the given function.
So, The correct option for the first part is
Option B -: The function is not increasing on any interval.
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