Help evaluating piecewise functions

[unf0r5ak3n]

Homework Statement

a) Write down the definition of "critical point." Find all critical points of ƒ(x).
b) Use the first derivative test to find the x-vale of which ƒ(x) has a minimum.
c) Is ƒ(x) differentiable at x = 7? Justify your answer.

Homework Equations

ƒ(x)={(0.3125(x-5)^2 + 1.3, x<7)
and (1.25x -6.25, x (> or equal to) 7)}

The Attempt at a Solution

Not sure how to do this, all I know is the definition of a critical point: points of a function where the function if not differentiable or its derivative is zero.

 

[mighty2000]

a) The definition of a critical point is a point on a function where the derivative is zero or undefined. In other words, it is a point where the function is not differentiable or where the slope of the tangent line is zero. To find the critical points of ƒ(x), we need to set the derivative of ƒ(x) equal to zero and solve for x.
ƒ'(x) = 0
0.625(x-5) = 0
x = 5
Therefore, x = 5 is the only critical point of ƒ(x).

b) To find the x-value where ƒ(x) has a minimum, we can use the first derivative test. This test states that if the derivative of a function changes from positive to negative at a point, then that point is a local minimum.
ƒ'(x) = 0.625(x-5)
For x < 5, ƒ'(x) is negative and for x > 5, ƒ'(x) is positive. Therefore, x = 5 is a local minimum of ƒ(x).

c) Yes, ƒ(x) is differentiable at x = 7. This is because the function is defined for all values of x, including x = 7. Additionally, the function is continuous at x = 7, which is a necessary condition for differentiability. Therefore, ƒ(x) is differentiable at x = 7.