AP EAMCET · Maths · Application of Derivatives

The curve $f (x) = e^{x} \sin x$ is defined in the interval $[0, 2 π] .$ The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum, is

A $\frac{π}{4}$
B $\frac{π}{2}$
C $\frac{π}{6}$
D $\frac{π}{3}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{π}{2}$

Step-by-step Solution

Detailed explanation

Given: f(x)=exsinx We know that the slope of tangent is given by dydx=f'x. Now, f'(x)=Z=exsinx+excosx Z=exsinx+cosx dZdx=exsinx+cosx+excosx-sinx ⇒dZdx=2excosx For maxima, dZdx=0 ⇒2excosx=0 ⇒excosx=0 ∵ex≠0, x∈R…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The curve f(x)=exsinx is defined in the interval [0,2π]. The value of x for which the slope of the tangent drawn to the curve at x is maximum, is

Step-by-step Solution

The curve $f (x) = e^{x} \sin x$ is defined in the interval $[0, 2 π] .$ The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum, is