평균값 정리 계산기
평균값 정리를 단계별로 적용하세요
이 계산기는 주어진 구간에서 주어진 함수에 대해 평균값 정리의 결론을 만족하는 모든 값 $$$c$$$를 (단계를 보여 주며) 찾아줍니다. 롤의 정리는 평균값 정리의 특수한 경우입니다( $$$f(a)=f(b)$$$인 경우).
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: $$$- \frac{10 \sqrt{3}}{3}\approx -5.77350269189626$$$, $$$\frac{10 \sqrt{3}}{3}\approx 5.77350269189626$$$