Question

Find the critical points of the following functions (in \( (-\infty, \infty) \) ), and determine whether they are local maximum points, local minimum points. or points of inflection:

(a) x² (x−1)²
(b) (4x+3)/(x² +1);
(c) 2x+log(1+x² );
(d) sin(x⁴);
(e) x/(a² +x ²),a=0
(f) coshx;
(g) xⁿ e⁻ˣ² (n positive integer);
(h) eˣ² + e⁻ˣ²

Answer 1

The question asks to identify critical points of various functions and classify them as local maxima, minima, or inflection points. This is typically done by finding the first and possibly second derivatives, setting the first derivative equal to zero, and analyzing the results to determine the nature of these points using concepts like the first derivative test and examining concavity.

Step-by-step explanation:

The task is to find critical points of given functions and determine if they are local maximum points, local minimum points, or points of inflection. The process involves taking the derivative of the function, setting it equal to zero, and solving for x to find critical points. Then, the second derivative is used to determine the nature of these points. Examples of more complex functions would require deeper analysis, such as using the first derivative test or concavity tests.

Each function provided in the query represents a unique case requiring separate differentiation and analysis. For instance, for a polynomial like x²(x−1)², one would find its first derivative, set it equal to zero, and then use test points to determine the nature of critical points. For a trigonometric function such as sin(x⁴), we would again differentiate and evaluate the function's slope changes around the critical points to ascertain their nature.