a. use the intermediate value theorem to show that the following equation has a solution on the given…
a. use the intermediate value theorem to show that the following equation has a solution on the given interval. b. use the graphing utility to find all the solutions to the equation on the given interval. c. illustrate your answers with an appropriate graph. the intermediate value theorem states that if f is continuous on the interval a,b and l is a number strictly between f(a) and f(b), then there exists at least one number c in (a,b) satisfying f(c)=l. for which value(s) of x is the function f(x)=\\sqrt{x^{4}+18x^{2}+5} continuous? a. it is continuous on 0,1, but not for all x b. it is continuous for some x but not on 0,1 c. it is continuous for all x d. it is not continuous on any interval
Answer
Explanation:
Step1: Recall continuity of a function
A function $y = f(x)=\sqrt{x^{4}+18x^{2}+5}$ is a composition of a square - root function and a polynomial function. The polynomial function $g(x)=x^{4}+18x^{2}+5$ is continuous for all real - valued $x$ since it is a polynomial. The square - root function $y = \sqrt{u}$ is continuous for $u\geq0$. Since $x^{4}+18x^{2}+5=(x^{2}+9)^{2}-76>0$ for all real $x$ (because $x^{2}\geq0$, so $x^{4}+18x^{2}+5\geq5>0$), the function $f(x)=\sqrt{x^{4}+18x^{2}+5}$ is continuous for all real $x$.
Answer:
C. It is continuous for all $x$