Question

Let $f\left(x\right)=e{}^x\sin x$ be the equation of a curve. If at $x=a,0\le a\le 2\pi$, the slope of the tangent is the maximum then the value of $a$ is

• A. $\pi /2$
• B. $3\pi /2$
• C. $\pi$
• D. $\pi /4$

Answer 1

The correct option is A $\pi /2$

Let $y=e^{x}sin\left(x\right)$
Slope is $\frac{dy}{dx}=e^x(sin\left(x\right)+cos\left(x\right))$
To maximize slope
$\frac{d^{2}y}{{dx}^2}={2e}^{x}cos\left(x\right)=0$
Critical point is $x=\frac{\pi }{2}$
And at $x=\frac{\pi }{2}$ is the point of maxima