How do you determine at which the graph of the function #y=x^2+1# has a horizontal tangent line?

Question

Charles Anctil · Accepted Answer

See below.

Horizontal tangents have a gradient of zero, so if we find the first derivative and solve it for zero, we can identify where these lie:

#d/dx(x^2+1)=2x#

#f'(x)=0#

#2x=0=>x=0 #

So horizontal tangent at #x= 0#:

#y=(0)^2+1=1#

coordinates #( 0 ,1)#

Equation of tangent line:

#y-1=0(x-0)#

#y=1#

Graph:

William Abdalla · Answer

undefined To determine at which point the graph of the function y=x^2+1 has a horizontal tangent line, we need to find the derivative of the function and set it equal to zero. Taking the derivative of y=x^2+1 with respect to x, we get dy/dx = 2x. Setting this derivative equal to zero, we have 2x = 0. Solving for x, we find x = 0. Therefore, the graph of the function y=x^2+1 has a horizontal tangent line at the point (0, 1).

Axel Alvarez · Answer

undefined To determine where the graph of the function $ y = x^2 + 1 $ has a horizontal tangent line, you need to find the points where the derivative of the function equals zero. So, find the derivative of the function $ y = x^2 + 1 $ with respect to $ x $, which is $ y' = 2x $. Then, set this derivative equal to zero and solve for $ x $ to find the critical points. After finding the critical points, you can determine whether the tangent lines are horizontal or not at those points by examining the slope of the tangent line.