SATHEE: Application of Derivatives 3 Question 1

Application of Derivatives 3 Question 1

1. If $f : R \to R$ is a twice differentiable function such that $f^{''} (x) > 0$ for all $x \in R$ , and $f \frac{1}{2} = \frac{1}{2}, f (1) = 1$ , then

(a) $f^{'} (1) \leq 0$

(b) $f^{'} (1) > 1$

(d) $\frac{1}{2} < f^{'} (1) \leq 1$

(2017 Adv.)

Assertion and Reason

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Answer:

Correct Answer: 1. (b)

Solution:

$f^{'} (x)$ is increasing

For some $x$ in $\frac{1}{2}, 1$

$f^{'} (x) = 1$

$∴ f^{'} (1) > 1$

[LMVT]

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Application of Derivatives 3 Question 1

1. If f:R→R is a twice differentiable function such that f′′(x)>0 for all x∈R, and f12=12,f(1)=1, then

Assertion and Reason

Answer:

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1. If $f : R \to R$ is a twice differentiable function such that $f^{''} (x) > 0$ for all $x \in R$ , and $f \frac{1}{2} = \frac{1}{2}, f (1) = 1$ , then