Show that the function f(t)=√t+√√1+t-4 has exactly one zero in Solve the equation √t+√√1+t-4 = 0 to find the zeros of the given √t+√1 +1 -4 = 0 √√1+t=4-√t 1+t=
Show that the function f(t)=√t+√√1+t-4 has exactly one zero in Solve the equation √t+√√1+t-4 = 0 to find the zeros of the given √t+√1 +1 -4 = 0 √√1+t=4-√t 1+t=
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Show that the function \( f(t) = \sqrt{t} + \sqrt{1+t} - 4 \) has exactly one zero in the interval \((0, \infty)\).
**Solution:**
Solve the equation:
\[
\sqrt{t} + \sqrt{1+t} - 4 = 0
\]
to find the zeros of the given function.
Rearranging the equation, we get:
\[
\sqrt{t} + \sqrt{1+t} = 4
\]
Let \( x = \sqrt{t} \), then \( t = x^2 \).
Substitute back into the equation:
\[
x + \sqrt{1+x^2} = 4
\]
Solving for \( x \), we further simplify:
\[
\sqrt{1+x^2} = 4 - x
\]
Squaring both sides, we obtain:
\[
1 + x^2 = (4-x)^2
\]
This equation can be solved for \( x \) to find the zero of the function in the given interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff17eecc8-d5a6-40f8-843b-f14b601b8c9d%2F7cdbfd53-0d9c-47d8-b1da-ed455f0d7da6%2Fq0e1cr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Show that the function \( f(t) = \sqrt{t} + \sqrt{1+t} - 4 \) has exactly one zero in the interval \((0, \infty)\).
**Solution:**
Solve the equation:
\[
\sqrt{t} + \sqrt{1+t} - 4 = 0
\]
to find the zeros of the given function.
Rearranging the equation, we get:
\[
\sqrt{t} + \sqrt{1+t} = 4
\]
Let \( x = \sqrt{t} \), then \( t = x^2 \).
Substitute back into the equation:
\[
x + \sqrt{1+x^2} = 4
\]
Solving for \( x \), we further simplify:
\[
\sqrt{1+x^2} = 4 - x
\]
Squaring both sides, we obtain:
\[
1 + x^2 = (4-x)^2
\]
This equation can be solved for \( x \) to find the zero of the function in the given interval.
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