Prove There is a real solution of an equation 5x x4 fx In

Prove There is a real solution of an equation 5^x = x^4 f(x) = In x is uniformly continuous on [1, infinity)

Solution

(a) It is easily identified that f(x)=5^x and g(x)=x⁴ are both continuous for all real values of x. Now, since we have to show that there is at least one real solution we check the value of both the functions at different points. We know that at x=0, f(x)=1 and g(x)=0 => f>g; Now g(x) is symmetric about x=0 and thus tends to infinity as x tends to -infinity, whereas f(x) tends to zero for x<<0, thus at some point f(x) becomes less than g(x) => there must be a crossing point which is the root for the equation.

(b) Suppose f:[1,)R has bounded derivative. Then f is uniformly continuous on its domain.

Here, f`(x)=1/x which is bounded in [1,) as it is continuously decreasing function from 1 to tending to zero.