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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2024 · Mathematics · Application of Derivatives

The non-negative values of b for which the function (16 x^3)/(3)-4...

The non-negative values of bb for which the function 16x33−4bx2+x\frac{16 x^3}{3}-4 b x^2+x has neither maximum nor minimum in the range x>0x > 0 is:

Options

  1. A0<b<10 < b < 1
  2. B1<b<21 < b < 2
  3. Cb>2b > 2
  4. D0≤b<10 \leq b < 1

Correct answer

D. 0≤b<10 \leq b < 1

Step-by-step solution

Let, f(x)=16x33−4bx2+xf(x)=16x2−8bx+1∵f(x)f(x)=\frac{16 x^3}{3}-4 b x^2+xf(x)=16 x^2-8 b x+1\because f(x) is neither maximum nor minimum then f(x)≠0f(x)>0 or f(x)<0\begin{aligned} f(x) & \neq 0 \\ f(x) & > 0 \text{ or } f(x) < 0 \end{aligned} For, f(x)>0\quad f(x) > 0D<0 and a>0D=64b2−64<0\begin{aligned} & \mathrm{D} < 0 \text{ and } a > 0 \\ & \mathrm{D}=64 b^2-64 < 0 \end{aligned}b2−1<0andbb^2-1 < 0 and b is non-negative as b≥0b \geq 0b∈[0,1]0≤b<1\begin{aligned} & & b & \in[0,1] \\ & & 0 & \leq b < 1 \end{aligned}$$
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