Maxima & Minima Calculator

Local maxima and minima, also called turning points, are where a graph stops rising and starts falling, or the other way round. There the derivative is zero. This calculator differentiates the function, solves f′(x) = 0 in a range, classifies each point with the second derivative test, finds points of inflection and graphs everything.

Try:

How to use the maxima & minima calculator

  1. Type the function, such as x^3 - 6x^2 + 9x + 1.
  2. Choose the range of x to search.
  3. Press Calculate to see the turning points and graph.

Formula

f′(x)=0f'(x) = 0
f′′(x)>0⇒minimum,f′′(x)<0⇒maximumf''(x) > 0 \Rightarrow \text{minimum},\quad f''(x) < 0 \Rightarrow \text{maximum}

Worked example: x³ − 6x² + 9x + 1

  • f(x) =: x^3 - 6x^2 + 9x + 1
  • Search x from: -10
  • to: 10

◐ Partly checked

local maximum: (1,  5)local minimum: (3,  1)\begin{gathered}\text{local maximum: } (1,\; 5)\\\text{local minimum: } (3,\; 1)\end{gathered}
f'(x)
3x2−12x+93x^{2} - 12x + 9
f''(x)
6x−126x - 12
Points of inflection
(2, 3)
−0.50.511.522.533.544.5−551015maxminf(x)f'(x)
Turning points of f (red) are where the dashed f'(x) crosses zero; green dots are points of inflection.
Step-by-step working (3 steps)

  1. Differentiate

    Find the first derivative.

    f′(x)=3x2−12x+9f'(x) = 3x^{2} - 12x + 9

  2. Solve f'(x) = 0

    Stationary points are where the slope is zero.

    x=1,  x=3x = 1,\; x = 3

  3. Second derivative test Second derivative test

    If f''(x) > 0 it is a minimum, if f''(x) < 0 a maximum.

    f′′(x)=6x−12f''(x) = 6x - 12
Formulas used
f′(x)=0f'(x) = 0
f′′(x)>0⇒min⁡,f′′(x)<0⇒max⁡f''(x) > 0 \Rightarrow \min,\quad f''(x) < 0 \Rightarrow \max
How this was checked
  • ✓ At each stationary point both the exact and a numerical derivative are 0.

Searches between -10 and 10; roots of f′ found numerically and rounded.

Frequently asked questions

What is a critical point?

A point where f′(x) = 0 (or is undefined). It may be a maximum, a minimum or a point of inflection.

How does the second derivative test work?

At a critical point, if f″ is positive the graph curves up (a minimum); if negative it curves down (a maximum).

What is a point of inflection?

Where the graph changes from curving up to curving down, so f″ changes sign.

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