Question

if f(a)=0 but f'(a) '0 then zeero is called of
order
O a. 1
O b. 3
O c. 2 2
O d. none​

Answer 1

Answer:

Step-by-step explanation:

Let f : I → R, I an interval. A point x0 ∈ I is a local maximum of f if there is

a δ > 0 such that f(x) ≤ f(x0) whenever x ∈ I ∩ (x0 − δ, x0 + δ). Similarly, we can define local

minimum.

Theorem 6.1 : Suppose f : [a, b] → R and suppose f has either a local maximum or a local

minimum at x0 ∈ (a, b). If f is differentiable at x0 then f

0

(x0) = 0.

Proof: Suppose f has a local maximum at x0 ∈ (a, b). For small (enough) h, f(x0 + h) ≤ f(x0).

If h > 0 then

f(x0 + h) − f(x0)

h

≤ 0.

Similarly, if h < 0, then

f(x0 + h) − f(x0)

h

≥ 0.

By elementary properties of the limit, it follows that f

0

(x0) = 0.