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if f(x)=(1+x)^r-(1+x^r) \ \ \text{where }x>0 \ and \ 0<r<1then Which of the following is true for f(x) ?

Option: 1

f(x) is a decreasing function


Option: 2

f(x) is a increasing function


Option: 3

can't say 


Option: 4

None of these 


Answers (1)

best_answer

 

 

Application of Monotonicity (Pat 2) -

Application of Monotonicity (Pat 2)

(c) Inequality Using Monotonicity

We can prove inequalities using the concept of monotonicity. To prove f(x) < g(x) in some interval, we consider another function such that h(x) = g(x) - f(x) or f(x) - g(x) and then we study the nature of h(x) w.r.t. Monotonicity of h(x) in the interval.

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f(x)=(1+x)^r-(1+x^r) \ \ \text{where }x>0 \ and \ 0<r<1 \\ f(x)=(1+x)^r-(1+x^r) \\ f'(x)=r(1+x)^{r-1}-rx^{r-1} \\ f'(x)=r[(1+x)^{r-1}-x^{r-1}]\\ 1+x>x\\ (1+x)^{1-r}>x^{1-r}\\ \frac{1}{(1+x)^{r-1}}> \frac{1}{x^{r-1}} \\ (1+x)^{r-1}<x^{r-1}\\ (1+x)^{r-1}-x^{r-1}<0\\ f'(x)<0 \\ \text{f(x) is decreasing function }\\ f(0)=0\\ f(x)<f(0) \ \ x>0\\

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Divya Prakash Singh

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