I need help with this question Show that the equation 5x x

I need help with this question: . Show that the equation 5^x = x^4 has at least one real solution

Solution

Examine the function f(x) = 3^x - x^2.
Find a value of f(x), for example the number \'c\', such that f(c) > 0.
Find a different value of f(x), for example the number \'d\', such that f(d) < 0

The graph of f(x) must cross the x-axis somewhere between c and d. (Or between d and c, depending on which is greater.) Therefore, f(x) = 0 for SOME value of x between c and d.

f(x) = 5^x - x^4

f(x) = 5^x - x^4 is continuous and f(-1) = -4/5 and f(0) = 1. Since these are of opposite sign f(x) must have a zero (by the intermediate value theorem) between -1 and 0 and so the equation has a solution in the interval [-1,0].