Applications of Derivatives Test

Result

Applications of Derivatives Test - 4

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Question 1
1 / -0

Let f (x) be a polynomial function of second degree. If f (1) = f (−1) and a, b, c are in A. P., then f′ (a), f′ (b) and f′ (c) are in

A
A.P.

B
G.P.

C
H. P.

D
arithmetic−geometric progression

Solution

f (x) = ax² + bx + c
f (1) = a + b + c
f (− 1) = a − b + c
⇒ a + b + c = a − b + c also 2b = a + c
f′ (x) = 2ax + b = 2ax
f′ (a) = 2a²
f′ (b) = 2ab
f′ (c) = 2ac
⇒ AP.
Question 2
1 / -0

Consider the function f(x) = |x – 2| + |x – 5|, x ∈ R.
Statement 1: f’(4) = 0
Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5)

A
Statement 1 is false, statement 2 is true

B
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

C
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

D
Statement 1 is true, statement 2 is false

Solution

f(x) = 7 – 2x; x < 2
= 3; 2≤ x ≤ 5
= 2x – 7; x > 5
f(x) is constant function in [2, 5]
f is continuous in [2, 5] and differentiable in (2, 5) and f(2) = f(5)
by Rolle’s theorem f’(4) = 0
∴ Statement 2 and statement 1 both are true and statement 2 is correct explanation for statement 1.
Hence, option B is correct.