For the function on L.H.S (a) cosx-1 +(x^2)/(2 !)-(x^3)/(3 !) (b) cosx-1+(x^2)/(2 !) (c) x^4 e^-

step 1 To find the extremum of a function, we find its derivative F dash of x and we equate that to 0.

For the function on L.H.S (a) cosx-1 +(x^2)/(2 !)-(x^3)/(3...

For the function on L.H.S (a) $\cos x-1$ $+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}$ (b) $\cos x-1+\frac{x^{2}}{2 !}$ (c) $x^{4} e^{-x^{2}}$ (d) $\sin 3 x-3 \sin x$ (p) minimum value is $-4$ (q) there is no extremum at $x=0$ (r) the minimum is $f(0)=0$ (s) the functions reaches maximum at $x=\sqrt{2}$

For the function on L.H.S
(a) $\cos x-1$
$+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}$
(b) $\cos x-1+\frac{x^{2}}{2 !}$
(c) $x^{4} e^{-x^{2}}$
(d) $\sin 3 x-3 \sin x$

(p) minimum value is $-4$
(q) there is no extremum at $x=0$
(r) the minimum is $f(0)=0$
(s) the functions reaches maximum at $x=\sqrt{2}$

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of that function at a single point. In this context, a Maclaurin series (which is a special case of the Taylor series around zero) is used to approximate trigonometric functions, helping to reveal the local behavior of the function close to that point. This approximation is key to understanding and analyzing the function’s overall behavior as a polynomial.

Critical Point Analysis

Critical point analysis involves finding where a function’s derivative is zero or undefined to identify candidates for local minima and maxima. This concept is essential as it underpins the process of locating and classifying extreme values in a function. By setting the derivative equal to zero and solving for the input values, one can determine where the function may exhibit a peak or a trough.

First and Second Derivative Tests

The first derivative test examines the sign changes of the derivative around critical points to determine whether a function is increasing or decreasing, hence identifying local extremes. The second derivative test further analyzes these points by assessing the concavity of the function, which indicates whether a point is a local minimum or maximum. Both tests provide systematic ways to classify critical points.

Exponential and Polynomial Function Behavior

Functions that combine polynomial and exponential factors, such as a polynomial multiplied by an exponential decay, exhibit unique behavior where the exponential term can dominate the long-term behavior. The interplay between the growth of the polynomial and the decay of the exponential term is crucial for understanding the global behavior of the function, especially in determining the existence and nature of extrema.

Trigonometric Function Properties

Trigonometric functions such as sine and cosine have inherent periodicity and symmetry, as well as well-established identities for sums, differences, and multiples of angles. These properties are vital when analyzing functions involving trigonometric terms, allowing for the simplification and transformation of expressions, which in turn facilitates the identification of their extrema and overall behavior.

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