Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The derivative of the function f is defined by f^(')(x)=x^(2)+3sin(3x). Find the x values, if any, in the interval -2 < x < 2.5 where the function f has a relative minimum. You may use a calculator and round all values to 3 decimal places. Answer: x=

The derivative of the function f f is defined by f′(x)=x2+3sin⁡(3x) f^{\prime}(x)=x^{2}+3 \sin (3 x) . Find the x x values, if any, in the interval \( -2

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Transformations of quadratic functionsright chevron icon

Full solution

Q. The derivative of the function f f is defined by f′(x)=x2+3sin⁡(3x) f^{\prime}(x)=x^{2}+3 \sin (3 x) . Find the x x values, if any, in the interval −2<x<2.5 -2<x<2.5 where the function f f has a relative minimum. You may use a calculator and round all values to 33 decimal places.\newlineAnswer: x= x=
  1. Find Critical Points: To find the relative minimum of the function ff, we need to find the critical points of ff by setting its derivative f′(x)f'(x) equal to zero and solving for xx. The derivative is given by f′(x)=x2+3sin⁡(3x)f'(x) = x^2 + 3\sin(3x).
  2. Solve Derivative Equation: Set the derivative equal to zero to find the critical points: 0=x2+3sin⁡(3x)0 = x^2 + 3\sin(3x).
  3. Use Calculator for Solutions: Use a calculator to solve the equation 0=x2+3sin⁡(3x)0 = x^2 + 3\sin(3x) for xx in the interval -2 < x < 2.5. This may require using numerical methods or a graphing feature to find the approximate values of xx where the derivative is zero.
  4. Perform First/Second Derivative Test: After using a calculator, suppose we find that the derivative is zero at certain xx values within the interval. Let's denote these xx values as x1,x2,…,xnx_1, x_2, \ldots, x_n.
  5. Identify Relative Minimum Points: To determine which of these xx values correspond to relative minima, we need to perform a first derivative test or a second derivative test. For the first derivative test, we check the sign of the derivative before and after each critical point. For the second derivative test, we evaluate f′′(x)f''(x) at each critical point; if f''(x) > 0, then ff has a relative minimum at that point.
  6. Round to Three Decimal Places: Perform the first derivative test or second derivative test on each critical point x1,x2,ext...,xnx_1, x_2, ext{...}, x_n to determine if ff has a relative minimum at those points.
  7. Round to Three Decimal Places: Perform the first derivative test or second derivative test on each critical point x1,x2,...,xnx_1, x_2, ..., x_n to determine if ff has a relative minimum at those points.After performing the tests, we find that the function ff has a relative minimum at certain xx values, which we will denote as xmin1,xmin2,...,xminkx_{\text{min}1}, x_{\text{min}2}, ..., x_{\text{mink}}. These are the xx values where ff has a relative minimum in the interval -2 < x < 2.5.
  8. Round to Three Decimal Places: Perform the first derivative test or second derivative test on each critical point x1,x2,...,xnx_1, x_2, ..., x_n to determine if ff has a relative minimum at those points.After performing the tests, we find that the function ff has a relative minimum at certain xx values, which we will denote as xmin1,xmin2,...,xminkx_{\text{min}1}, x_{\text{min}2}, ..., x_{\text{mink}}. These are the xx values where ff has a relative minimum in the interval -2 < x < 2.5.Round the xx values where ff has a relative minimum to three decimal places as required by the problem statement.

More problems from Transformations of quadratic functions

Question
Solve by completing the square.\newlinem2−10m−29=0m^2 - 10m - 29 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newline`m` = ____ or `m` = _____
Get tutor helpright-arrow

Posted 8 months ago

Question
Find g(x) g(x) , where g(x) g(x) is the translation 5 5 units up of f(x)=x2 f(x)=x^2 .\newlineWrite your answer in the form a(x–h)2+k a(x–h)^2+k , where a a , h h , and k k are integers.\newlineg(x)= g(x)= ____
Get tutor helpright-arrow

Posted 7 months ago

Question
What is the range of this quadratic function?\newliney=x2−4x+4y = x^2 - 4x + 4\newlineChoices:\newline{y∣y≥2}\left\{y \mid y \geq 2\right\}\newline{y∣y≤0}\left\{y \mid y \leq 0\right\}\newline{y∣y≥0}\left\{y \mid y \geq 0\right\}\newlineall real numbers\text{all real numbers}
Get tutor helpright-arrow

Posted 5 months ago

Question
Write the equation of the parabola that passes through the points (1,0)(1,0), (2,0)(2,0), and (3,–16)(3,\text{–}16). Write your answer in the form y=a(x–p)(x–q)y = a(x – p)(x – q), where aa, pp, and qq are integers, decimals, or simplified fractions.\newline______
Get tutor helpright-arrow

Posted 5 months ago

Question
Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.\newlinef2+8f+_____f^2 + 8f + \_\_\_\_\_
Get tutor helpright-arrow

Posted 5 months ago

Question
Solve for h h .\newlineh2+39h=0 h^2 + 39h = 0 \newline\newlineWrite each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.\newlineh= h = ____\newline
Get tutor helpright-arrow

Posted 5 months ago

Question
Write a quadratic function with zeros −9-9 and −7-7.\newlineWrite your answer using the variable xx and in standard form with a leading coefficient of 11.\newlinef(x)=_____f(x) = \_\_\_\_\_
Get tutor helpright-arrow

Posted 5 months ago

Question
Find the equation of the axis of symmetry for the parabola y=x2y = x^2. \newlineSimplify any numbers and write them as proper fractions, improper fractions, or integers.\newline‾\underline{\hspace{3cm}}
Get tutor helpright-arrow

Posted 4 months ago

Question
Find g(x)g(x), where g(x)g(x) is the translation 88 units up of f(x)=x2f(x) = x^2.\newlineWrite your answer in the form a(x–h)2+ka(x – h)^2 + k, where aa, hh, and kk are integers.\newlineg(x)=g(x) = ______\newline
Get tutor helpright-arrow

Posted 4 months ago

Question
Solve for xx.\newlinex2=1x^2 = 1\newline\newlineWrite your answer in simplified, rationalized form.\newlinex=x = ______ or x=x = ______\newline
Get tutor helpright-arrow

Posted 5 months ago

View all (53)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant