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Sahithyan's S1
Sahithyan's S1 — Mathematics

Intermediate Value Theorem for Integrals

Suppose ff is a continuous function on [a,b][a,b]. Then x(a,b)\exists x \in (a,b):

f(x)=1baabff(x)=\frac{1}{b-a}\int_a^b f

Proof

Suppose fmax=M=f(x0)f_{\text{max}} = M = f(x_0) and fmin=m=f(y0)f_\text{min} = m=f(y_0).

When M=mM=m: ff is a constant function. Proof is trivial.

Otherwise:

m(ba)abfM(ba)m(b-a) \le \int_a^b f \le M(b-a)

Then there exists x(x0,y0)x \in (x_0, y_0).