JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f(x) = e^x -x\) and \(g(x) = x^2 -x\), \(\forall \in R\). Then the set of all \(x \in R\), where the function \(h(x) = (fog)\, (x)\) is increasing is
- A \(\left[ {0,\frac{1}{2}} \right] \cup \left[ {1,\infty } \right)\)
- B \(\left[ {1,\frac{1}{2}} \right] \cup \left[ {\frac{1}{2},\infty } \right)\)
- C \(\left[ {\frac{{ - 1}}{2},0} \right] \cup \left[ {1,\infty } \right)\)
- D \(\left[ {0,\infty } \right)\)
Answer & Solution
Correct Answer
(A) \(\left[ {0,\frac{1}{2}} \right] \cup \left[ {1,\infty } \right)\)
Step-by-step Solution
Detailed explanation
\(h(x)=f(g(x))\) \(\therefore h^{\prime}(x)=f^{\prime}(g(x)) g^{\prime}(x)\) and \(f^{\prime}(x)=e^{x}-1\) \({h^\prime }(x) = \left( {{e^{g(x)}} - 1} \right){g^\prime }(x)\) \(h^{\prime}(x)=\left(e^{x^{2}-x}-1\right)(2 x-1) \geq 0\) case: 1 \(e^{x^{2}-x} \leq 1\) and…
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