How do you determine the intervals where #f(x)=3x-4# is concave up or down?

Question

Emily Ammerman · Accepted Answer

#f(x) = 3x-4# is never concave up or concave down.

By definition, a function #f(x)# is concave up when #f''(x) > 0#, and it is concave down when #f''(x) < 0#. Let #f(x) = 3x - 4#. #f'(x) = 3# #f''(x) = 0# Here, we notice that the second derivative is never greater than or less than 0, which means #f(x) = 3x-4# is never concave up or concave down.

Isaiah Allen · Answer

Neither- point of inflection

When we want to determine if a function is concave up or concave down, we want to analyze the function's second derivatives #f'(x)=3# #f''(x)=0# (Derivative of a constant is zero) We have three possible scenarios: #f''(x)>0=>#Function is concave up #f''(x)<0=>#Function is concave down #f''(x)=0=>#Point of inflection (neither concave up or down) We see that our second derivative of #f(x)# is zero, which means we are in scenario three: #f(x)# is neither concave up nor down...we have a point of inflection. Hope this helps!

Emilia Aguilar · Answer

undefined To determine the intervals where $ f(x) = 3x - 4 $ is concave up or down, you need to find the second derivative of the function. If the second derivative is positive, the function is concave up on that interval. If the second derivative is negative, the function is concave down on that interval.

The second derivative of $ f(x) = 3x - 4 $ is $ f''(x) = 0 $. Since the second derivative is constant and equal to zero, it means that the concavity of the function does not change. Thus, $ f(x) = 3x - 4 $ is neither concave up nor concave down; it is linear.