Which is an irrational number?

A. is the answer it can't be rational

Step-by-step explanation:

√21 is irrational.

Let's assume that √21 is rational. So √21 can be expressed in the form p/q form.

p/q is the reduced form of rational number so p and q have no common factors other than 1, i.e. they are co-prime numbers.

√21 = p/q

21 = p^2/q^2... (squaring both the sides)

21q^2 = p^2

p^2 is even. p^2 is a factor of 2...(1)

p=2t p^2=4t^2 t belongs to I

2q^2 = 4t^2 (2q^2=p^2)

q^2 = 2t^2

q^2 is even

q is even

2 is a factor of q.

From the statements (1) and (2), 2 is a common factor of p and q both.

This is contradictory because in p/q, we have assumed that p and q have no common factors other than 1.

That is, our assumption that √21 is rational is wrong.

Therefore, √21 is irrational.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.