Chapter 3: Q. 95 (page 277)

Prove that if $f''$ is zero on an interval, then f is linear on that interval.

Short Answer

Expert verified

$f (x) = a x + b .$ Hence, proved.

Step by step solution

Step 1. Given Information.

Second derivative of the function is zero on an interval.

Step 2. Theorem

The Derivative Measures Where a Function is Increasing or Decreasing

Let f be a function that is differentiable on an interval I.

(a) If $f'$ is positive in the interior of I, then f is increasing on I.

(b) If $f'$ is negative in the interior of I, then f is decreasing on I.

Step 3. Proof

$f'' (x) = 0, I n t e g r a t i n g w . r . t x w e g e t, f' (x) = a, w h e r e a i s a c o n s \tan t . I n t e g r a t i n g w . r . t x o n c e a g a i n w e g e t, f (x) = a x + b, w h e r e a a n d b b o t h a r e c o n s \tan t s .$

Since $f''$ is zero it is sure that $f'$ will be a constant and if first derivative is a constant then the function will be a linear function. Hence, Proved that the function will be linear.

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Prove that if $f''$ is zero on an interval, then f is linear on that interval.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Theorem

Step 3. Proof

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Prove that iff'' is zero on an interval, then f is linear on that interval.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Theorem

Step 3. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

Prove that if $f''$ is zero on an interval, then f is linear on that interval.