If f(x)=1+(x^2)/(2 !)+(x^4)/(4 !)+…∞and g(x)=x+(x^3)/(3 !)+(x^5)/(5 !)+…∞and ϕ(x)=f(x)+g(x), the

step 1 For solving the problem, fx equal to 1 plus x squared over 2 factorial plus x x x .m. 4 over 4 f

If f(x)=1+(x^2)/(2 !)+(x^4)/(4 !)+…∞and g(x)=x+(x^3)/(3...

If $\mathrm{f}(\mathrm{x})=1+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{4}}{4 !}+\ldots \infty$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}+\frac{\mathrm{x}^{3}}{3 !}+\frac{\mathrm{x}^{5}}{5 !}+\ldots \infty$ and $\phi(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})$, then $\phi(\mathrm{x})$ is (a) an increasing function for all positive real values of $x$ and decreasing for negative real values of $x$. (b) an increasing function for all real values of $x$. (c) a decreasing function for all real values of $x$. (d) a decreasing function for all positive real values of $x$ and increasing for negative real values of $x$.

If $\mathrm{f}(\mathrm{x})=1+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{4}}{4 !}+\ldots \infty$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}+\frac{\mathrm{x}^{3}}{3 !}+\frac{\mathrm{x}^{5}}{5 !}+\ldots \infty$ and $\phi(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})$, then $\phi(\mathrm{x})$ is
(a) an increasing function for all positive real values of $x$ and decreasing for negative real values of $x$.
(b) an increasing function for all real values of $x$.
(c) a decreasing function for all real values of $x$.
(d) a decreasing function for all positive real values of $x$ and increasing for negative real values of $x$.

Key Concepts

Maclaurin Series

A Maclaurin series is a special case of the Taylor series, representing a function as an infinite sum of its derivatives at zero. This series expansion is particularly useful for functions such as exponentials, trigonometric functions, and hyperbolic functions, allowing for approximation and analysis of their behavior over their entire domain.

Hyperbolic Functions

Hyperbolic functions, including hyperbolic cosine (cosh) and hyperbolic sine (sinh), are analogues of the trigonometric functions but are defined using exponential functions. Their Taylor series expansions demonstrate distinctive patterns, with cosh composed of even-powered terms and sinh of odd-powered terms, which helps relate them directly to the exponential function.

Exponential Function

The exponential function, denoted as e^x, is a fundamental mathematical function known for its property of being equal to its own derivative. It can be represented as an infinite series, and due to its strictly increasing nature over the entire set of real numbers, it serves as a key tool in understanding growth processes and the behavior of related function series.

Monotonicity

Monotonicity refers to a function's behavior in terms of consistently increasing or decreasing. When a function's derivative is positive throughout its domain, the function is said to be strictly increasing. This concept is crucial for analyzing functions represented by series, as it allows conclusions to be drawn about their overall trend based on their derivative behavior.

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