Question

Let f(x)=2x² −8x+5. a) Find the critical point c of f and compute f(c).

A) c=2, f(c)=−3

B) c=4, f(c)=5

C) c=1, f(c)=9

D) c=3, f(c)=10

Answer 1

The critical point c of the function f(x) = 2x² - 8x + 5 is found by setting its derivative, 4x - 8, equal to zero. Solving for x gives x = 2, which is the critical point c. After substituting c into the original function, we get f(c) = -3, which corresponds to answer choice A.

Step-by-step explanation:

To find the critical point c of the function f(x) = 2x² − 8x + 5, we need to find the value of x at which the derivative of f(x) is equal to zero. This is because critical points occur where the derivative is zero or undefined. The critical points indicate where the function might have a local maximum or minimum, or a point of inflection.

The first step is to take the derivative of f(x). The derivative of f(x) with respect to x is f'(x) = 4x - 8. We set this equal to zero and solve for x to find the critical point:

f'(x) = 4x - 8 = 0
4x = 8
x = 2

The critical point c is therefore 2. To find f(c), we plug c into the original function:

(2) = 2(2)² - 8(2) + 5
= 8 - 16 + 5
= -3

So, the correct answer is A) c=2, f(c)=-3.