Consider the function f(x)=e^x-x^3 a) Find f^'(x) and f^''(x) b) Find the x -coordinates (accura

step 1 Solution A is FX is equal to E to x minus x cubed, F double dashed X is equal to E to x minus 6x

Consider the function f(x)=e^x-x^3 a) Find f^'(x) and...

Consider the function $f(x)=e^{x}-x^{3}$ a) Find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ b) Find the $x$ -coordinates (accurate to three significant figures) for any points where $f^{\prime}(x)=0$ c) Indicate the intervals for which $f(x)$ is increasing, and indicate the intervals for which $f(x)$ is decreasing. d) For the values of $x$ found in part $b$ ), state whether that point on the graph of $f$ is a maximum, minimum or neither. e) Find the $x$ -coordinate of any inflexion point(s) for the graph of $f$ f) Indicate the intervals for which $f(x)$ is concave up, and indicate the intervals for which $f(x)$ is concave down.

Consider the function $f(x)=e^{x}-x^{3}$
a) Find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$
b) Find the $x$ -coordinates (accurate to three significant figures) for any points where $f^{\prime}(x)=0$
c) Indicate the intervals for which $f(x)$ is increasing, and indicate the intervals for which $f(x)$ is decreasing.
d) For the values of $x$ found in part $b$ ), state whether that point on the graph of $f$
is a maximum, minimum or neither.
e) Find the $x$ -coordinate of any inflexion point(s) for the graph of $f$
f) Indicate the intervals for which $f(x)$ is concave up, and indicate the intervals for which $f(x)$ is concave down.

Key Concepts

Concavity and Inflection Points

Concavity describes the curvature of a function’s graph. A function is concave up when its graph lies above its tangent lines (indicating a positive second derivative) and concave down when it lies below (indicating a negative second derivative). An inflection point occurs where the concavity changes from up to down or vice versa, which is identified by a change in the sign of the second derivative.

Local Extrema Analysis

Local extrema, which include local maxima and minima, are determined by analyzing the critical points with tests such as the first derivative test or the second derivative test. These tests help in establishing whether a critical point corresponds to a maximum (where the function switches from increasing to decreasing) or a minimum (where the function switches from decreasing to increasing), or if it is neither.

Intervals of Increasing and Decreasing Functions

By analyzing the sign of the first derivative, one can determine the intervals where a function is increasing (when the first derivative is positive) or decreasing (when the first derivative is negative). This analysis is crucial for understanding the overall behavior of the function and for locating the extrema on its graph.

Critical Points

Critical points occur where the first derivative is equal to zero or is undefined. These points are important because they are potential locations for local maxima, local minima, or saddle points. Analyzing the sign of the derivative around these points helps determine the nature of these critical points, such as whether they represent a maximum, minimum, or neither.

Second Derivative

The second derivative of a function is the derivative of the first derivative and provides information about the curvature or concavity of the function. It helps in identifying inflection points, which are points where the function changes its concavity. Additionally, the sign of the second derivative indicates whether the graph of the function is concave up (positive second derivative) or concave down (negative second derivative).

Derivative

The derivative of a function represents its instantaneous rate of change and is foundational in determining the behavior of a function. It is calculated using derivative rules such as the sum rule, product rule, chain rule, and power rule. In the context of the given function, the first derivative is used to identify points where the function's slope is zero or undefined, which are potential candidates for local maxima or minima.

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