Between any two real roots of the equation $e^{x} s in x = 1$ , the equation $e^{x} cos x = - 1$ has

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — at least one root

Detailed Solution

$e^{x} s in x = 1 \Leftrightarrow g (x) = e^{- x} - s in x = 0$ .

If $α < β$ are two roots, then $g (α) = g (β) = 0$ and $g$ is differentiable, so by Rolle's theorem $g' (c) = 0$ for some $c$ in $(α, β)$ .

$g' (x) = - e^{- x} - cos x = 0 \Leftrightarrow e^{x} cos x = - 1$ . So there is at least one root between them.

✓ Check

e^{x} s in x = 1

has roots near

0.589

and

3.096

;

e^{x} cos x = - 1

has a root near

1.746

, which lies between them.

⚠ Common trap

(B) fails for roots that are not next to each other: between

0.589

and

6.285

there are two roots of

e^{x} cos x = - 1

(near

1.746

and

4.703

Quick tip

Rolle's theorem on $g (x) = e^{- x} - sin x$ : $g^{'} (x) = 0 ⟺ e^{x} cos x = - 1$

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Applications of Derivatives — Preparation

This question tests your understanding of Applications of Derivatives, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

This topic uses derivatives to solve practical and geometric problems. It covers the equations of tangents and normals to a curve, the rate of change of quantities (such as how fast the area of a circle grows as its radius increases), approximations using differentials, and increasing and decreasing functions. It also covers maxima and minima, using the first derivative test and the second derivative test, and word problems on optimisation such as finding the largest area for a fixed perimeter. Rolle's theorem and the mean value theorem are included as well. For maxima and minima, set f'(x) = 0 to find the critical points, then check f''(x): a negative value means a maximum and a positive value means a minimum. Remember that the slope of the tangent at a point is f'(x) and the slope of the normal is -1/f'(x). Practise the applications of derivatives questions below, each with a step-by-step solution.

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