Calculatrice du théorème de la valeur moyenne

La calculatrice trouvera tous les nombres $$$c$$$ (avec les étapes indiquées) qui satisfont aux conclusions du théorème de la valeur moyenne pour la fonction donnée sur l'intervalle donné. Le théorème de Rolle est un cas particulier du théorème de la valeur moyenne (lorsque $$$f(a)=f(b)$$$).

Enter a function:

Enter an interval: $$$[$$$, $$$]$$$

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Solution

Your input: find all numbers $$$c$$$ (with steps shown) to satisfy the conclusions of the Mean Value Theorem for the function $$$f=x^{3} - 2 x$$$ on the interval $$$\left[-10, 10\right]$$$.

The Mean Value Theorem states that for a continuous and differentiable function $$$f(x)$$$ on the interval $$$[a,b]$$$ there exists such number $$$c$$$ from the interval $$$(a,b)$$$, that $$$f'(c)=\frac{f(b)-f(a)}{b-a}$$$.

First, evaluate the function at the endpoints of the interval:

$$$f \left( 10 \right) = 980$$$

$$$f \left( -10 \right) = -980$$$

Next, find the derivative: $$$f'(c)=3 c^{2} - 2$$$ (for steps, see derivative calculator).

Form the equation: $$$3 c^{2} - 2=\frac{\left( 980\right)-\left( -980\right)}{\left( 10\right)-\left( -10\right)}$$$

Simplify: $$$3 c^{2} - 2=98$$$

Solve the equation on the given interval: $$$c=- \frac{10 \sqrt{3}}{3}$$$, $$$c=\frac{10 \sqrt{3}}{3}$$$

Answer: $$$- \frac{10 \sqrt{3}}{3}\approx -5.77350269189626$$$, $$$\frac{10 \sqrt{3}}{3}\approx 5.77350269189626$$$