Essentials of Calculus use a calculator to estimate any extrema

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SUMMARY

This discussion focuses on estimating extrema of the function f(x) = 5x³ - 30x² + 45x + 5√x using a calculator. Participants suggest using a graphing calculator to visualize the function and its derivative, f'(x) = 15x² - 60x + 45 + (5/2)x^(-1/2), to identify where it crosses the x-axis. Additionally, methods such as Newton's method are recommended for more precise calculations. The emphasis is on graphical analysis to locate maxima, minima, and inflection points.

PREREQUISITES
  • Understanding of calculus concepts, particularly derivatives
  • Familiarity with graphing calculators
  • Knowledge of Newton's method for numerical solutions
  • Basic algebra skills for manipulating polynomial functions
NEXT STEPS
  • Learn how to use a graphing calculator effectively for function analysis
  • Study Newton's method for finding roots of equations
  • Explore graphical techniques for identifying extrema and inflection points
  • Review the implications of the first and second derivative tests in calculus
USEFUL FOR

Students and educators in calculus, particularly those seeking to understand how to estimate function extrema using calculators and graphical methods.

Nawz
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Homework Statement




Use a calculator to estimate any extrema of this function:
f(x)=5x^3-30x^2+45x+5(sqrt(x))

Homework Equations





The Attempt at a Solution



I don't know how to find it using a calculator.

f'(x)= 15x^2-60x+45+(5/2)x^(-1/2)
 
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If your calculator is a graphing calculator, graph y= f'(x) and see where it crosses the x- axis. Another method of solving such an equation is to use a "Newton's method" or some other numerical method.
 
Nawz said:

Homework Statement




Use a calculator to estimate any extrema of this function:
f(x)=5x^3-30x^2+45x+5(sqrt(x))

Homework Equations





The Attempt at a Solution



I don't know how to find it using a calculator.

f'(x)= 15x^2-60x+45+(5/2)x^(-1/2)

Since you are estimating, you can simply graph f(x) and look for peaks (maxes), troughs (mins), and where the graph changes concavity (inflection points; though most books do not consider this an extremum). The derivative is not really needed, unless you want a more accurate estimate.
 

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