A curve y=f(x) passing through the point (1, (1/√e)) satisfies the differential equation (d y/d x)+x-(x2/2)=0. Then which of the following does not hold good?

Question

A curve y=f(x) passing through the point (1, (1/√e)) satisfies the differential equation (d y/d x)+x-(x2/2)=0. Then which of the following does not hold good? (A) f(x) is differentiable at x=0. (B) f(x) is symmetric w.r.t. the origin. (C) f(x) is increasing for x<0 and decreasing for x >0. (D) f ( x ) has two inflection points.

Tardigrade Admin · Accepted Answer

y=f(x)=e-(x2/2). The graph is as shown

Q. A curve $y = f (x)$ passing through the point $(1, \frac{1}{e})$ satisfies the differential equation $\frac{d y}{d x} + x^{- \frac{x ^{2}}{2}} = 0$ . Then which of the following does not hold good?

$f (x)$ is differentiable at $x = 0$ .

$f (x)$ is symmetric w.r.t. the origin.

$f (x)$ is increasing for $x < 0$ and decreasing for $x > 0$ .

$f (x)$ has two inflection points.

$y = f (x) = e^{- \frac{x ^{2}}{2}}$ . The graph is as shown

Q. A curve y=f(x) passing through the point (1,e​1​) satisfies the differential equation dxdy​+x−2x2​=0. Then which of the following does not hold good?

f(x) is differentiable at x=0.

f(x) is symmetric w.r.t. the origin.

f(x) is increasing for x<0 and decreasing for x>0.

f(x) has two inflection points.