JEE Mains · Maths · STD 12 - 6. Application of derivatives
The number of real roots of the equation \(\mathrm{e}^{4 \mathrm{x}}+2 \mathrm{e}^{3 \mathrm{x}}-\mathrm{e}^{\mathrm{x}}-6=0\) is :
- A \(2\)
- B \(4\)
- C \(1\)
- D \(0\)
Answer & Solution
Correct Answer
(C) \(1\)
Step-by-step Solution
Detailed explanation
Let \(\mathrm{e}^{\mathrm{x}}=\mathrm{t}>0\) \(f(t)=t^{4}+2 t^{3}-t-6=0\) \(f^{\prime}(t)=4 t^{3}+6 t^{2}-1\) \(f^{\prime \prime}(\mathrm{t})=12 \mathrm{t}^{2}+12 \mathrm{t}>0\) \(f(0)=-6, f(1)=-4, f(2)=24\) \(\Rightarrow\) Number of real roots \(=1\)
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