Let k and K be the minimum and the maximum va | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $f(x) = \frac{(1 + x)^{3/5}}{1 + x^{3/5}}$ for $x \in [0, 1]$.Taking the derivative $f'(x)$:$f'(x) = \frac{(1 + x^{3/5}) \cdot \frac{3}{5}(1 +

Vedclass · Accepted Answer

$(2^{-0.4}, 1)$

Vedclass · Answer

(A) Let $f(x) = \frac{(1 + x)^{3/5}}{1 + x^{3/5}}$ for $x \in [0, 1]$.Taking the derivative $f'(x)$:$f'(x) = \frac{(1 + x^{3/5}) \cdot \frac{3}{5}(1 + x)^{-2/5} - (1 + x)^{3/5} \cdot \frac{3}{5}x^{-2/5}}{(1 + x^{3/5})^2}$$f'(x) = \frac{3}{5} \left[ \frac{1 + x^{3/5}}{(1 + x)^{2/5}} - \frac{(1 + x)^{3/5}}{x^{2/5}} \right]$$f'(x) = \frac{3}{5} \left[ \frac{x^{2/5}(1 + x^{3/5}) - (1 + x)^{3/5}(1 + x)^{2/5}}{x^{2/5}(1 + x)^{2/5}} \right]$$f'(x) = \frac{3}{5} \left[ \frac{x^{2/5} + x - (1 + x)}{x^{2/5}(1 + x)^{2/5}} \right] = \frac{3}{5} \left[ \frac{x^{2/5} - 1}{x^{2/5}(1 + x)^{2/5}} \right]$Since $x \in [0, 1]$,$x^{2/5} \le 1$,so $f'(x) \le 0$. The function is monotonically decreasing.Thus,the maximum value $K = f(0) = \frac{(1+0)^{0.6}}{1+0^{0.6}} = 1$.The minimum value $k = f(1) = \frac{(1+1)^{0.6}}{1+1^{0.6}} = \frac{2^{0.6}}{2} = 2^{0.6 - 1} = 2^{-0.4}$.Therefore,the ordered pair $(k, K)$ is $(2^{-0.4}, 1)$.

Let $k$ and $K$ be the minimum and the maximum values of the function $f(x) = \frac{(1 + x)^{0.6}}{1 + x^{0.6}}$ in $[0, 1]$ respectively,then the ordered pair $(k, K)$ is equal to

Similar Questions

If the set of all values of $a$,for which the equation $5x^3 - 15x - a = 0$ has three distinct real roots,is the interval $(\alpha, \beta)$,then $\beta - 2\alpha$ is equal to . . . . . . .

The curve $f(x) = e^x \sin x$ is defined in the interval $[0, 2 \pi]$. The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum,is

The maximum area of a parallelogram inscribed in a triangle $ABC$ (given that $A$ is also one of the vertices of the parallelogram) is equal to (where symbols have their usual meaning in $\Delta ABC$):

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$,then the ordered pair $(m, M)$ is equal to

What are the minimum and maximum values of the function $f(x) = x^5 - 5x^4 + 5x^3 - 10$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module