Rolle's Theorem & Second Derivative Test Question

Question

Okay, so when going over some problems to prepare for a test that is coming up soon, I have these two questions im not sure on. I know these types of questions will be on the test, so I need help understanding how to solve and the step by step process.

 

The first one:

Determine if Rolle's Theorem can be applied to f on [1,2]. If so, find all values of c in the interval for which f'(c)=0

f(x) = x²-3x+2

 

The second one:

Use the second derivative test to locate all relative extrema and inflection points of f(x)=2x⁴-8x+3

Answer 1

Rolle's Theorem is applicable if f(x) is a continuous function in [a,b] and differentiable everywhere between (a,b), and f(a) = f(b).

f(x) = x²-3x+2      [1,2]

Here a = 1, b =2

f(x) is a continuous function and differentiable within (1,2). Also, f(1) = f(2)  check this.

Hence, Rolle's Theorem is applicable.

 

Next, find f'(x).   Set  f'(c) = 0 and solve for c

 

Second Problem.

Find f'(x).  Set this equal to zero to find the turning points. The solution reveals 1 turning point - say c.

Find f"(x).

What is f"(c)? Since f"(c) = a positive value, the turning point is a local minimum.

To find the points of inflection, set f"(x) = 0 and solve for x.