# Mean Value Theorem Calculator

The calculator will find all numbers c (with steps shown) that satisfy the conclusions of the Mean Value Theorem for the given function on the given interval.

• In general, you can skip the multiplication sign, so 5x is equivalent to 5*x.
• In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x).
• Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)).
• If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
• Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x).
• Similarly, tanxsec^3x will be parsed as tan(xsec^3(x)). To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x).
• From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x).
• If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
• All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
 Type Get Constants e e pi pi i i (imaginary unit) Operations a+b a+b a-b a-b a*b a*b a^b, a**b a^b sqrt(x), x^(1/2) sqrt(x) cbrt(x), x^(1/3) root(3)(x) root(x,n), x^(1/n) root(n)(x) x^(a/b) x^(a/b) abs(x) |x| Functions e^x e^x ln(x), log(x) ln(x) ln(x)/ln(a) log_a(x) Trigonometric Functions sin(x) sin(x) cos(x) cos(x) tan(x) tan(x), tg(x) cot(x) cot(x), ctg(x) sec(x) sec(x) csc(x) csc(x), cosec(x) Inverse Trigonometric Functions asin(x), arcsin(x), sin^-1(x) asin(x) acos(x), arccos(x), cos^-1(x) acos(x) atan(x), arctan(x), tan^-1(x) atan(x) acot(x), arccot(x), cot^-1(x) acot(x) asec(x), arcsec(x), sec^-1(x) asec(x) acsc(x), arccsc(x), csc^-1(x) acsc(x) Hyperbolic Functions sinh(x) sinh(x) cosh(x) cosh(x) tanh(x) tanh(x) coth(x) coth(x) 1/cosh(x) sech(x) 1/sinh(x) csch(x) Inverse Hyperbolic Functions asinh(x), arcsinh(x), sinh^-1(x) asinh(x) acosh(x), arccosh(x), cosh^-1(x) acosh(x) atanh(x), arctanh(x), tanh^-1(x) atanh(x) acoth(x), arccoth(x), cot^-1(x) acoth(x) acosh(1/x) asech(x) asinh(1/x) acsch(x)

Enter a function:

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

Write all suggestions in comments below.

## 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 continuous and differentiable function $$f(x)$$$ on the interval $$[a,b]$$$there exists such number $$c$$$ from that interval, 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 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: $$- \frac{10 \sqrt{3}}{3}$$$, $$\frac{10 \sqrt{3}}{3}$$$.