Question

The values of \( a \), if \( f(x) = 2e^x - a e^{-x} + (2a+1)x - 3 \) increases for all \( x \), are in

• A. \([0,\infty)\)
• B. \((-\infty,0]\)
• C. \((-\infty,\infty)\)
• D. \((1,\infty)\)

Hint:

For expressions like \(e^x\) and \(e^{-x}\), substitute \(t=e^x\) and apply AM-GM.

Answer 1

The Correct Option is A

Concept: Function increases for all \(x\) if: \[ f'(x) \ge 0 \quad \forall x \]

Step 1: Differentiate.
\[ f'(x) = 2e^x + ae^{-x} + (2a+1) \]

Step 2: Let \(t=e^x>0\).
\[ f'(x) = 2t + \frac{a}{t} + (2a+1) \]

Step 3: Minimize expression.
Using AM-GM: \[ 2t + \frac{a}{t} \ge 2\sqrt{2a} \quad \text{(valid when } a \ge 0\text{)} \] \[ \Rightarrow f'(x) \ge 2\sqrt{2a} + (2a+1) \]

Step 4: Ensure non-negative.
For \(a \ge 0\), RHS is always positive. \[ \Rightarrow f'(x) \ge 0 \quad \forall x \] Conclusion: \[ {a \in [0,\infty)} \]