Q. 2 TB - Calculus

Question

Critical points: What is the definition of a critical point for a function of a single variable? How do we use critical points to locate the extrema of the function?

Step-by-Step Solution

Answer

The critical point of a single variable function is the value $$x$$ in the domain of a function of a single real variable, $$f(x)$$, where it is not differentiable or when its derivative is zero.

The critical points can be used to locate the extrema of the function by evaluating the sign of $$f(x)$$ in the interval between the critical points and analyzing the sign change of the derivatives.

Step 1. Given Information

Critical point for a function of a single variable 

Step 2. Explanation

The critical point of a function is a value $$x$$ in the domain of a function of a single real variable, $$f(x)$$, that is not differentiable or has a zero derivative.

Here, $$f^{'}(x)=0$$

We can use critical points to locate the extrema of the function as follows:

Firstly, equate the first derivative of $$f(x)$$ to zero and find its critical points.

Next, find the intervals between the critical points and check whether $$f^{'}(x)$$ is positive or negative in that interval.

Find the critical points which changes the sign of the derivatives obtained as  local extrema and also ignore the values of $$x$$ that are in the domain of $$f(x)$$. If the derivative sign changes from positive to negative, then $$f(x)$$ has a local maximum and if the derivative sign changes from negative to positive, then $$f(x)$$ has a local minimum.