Calculus
# Curve Sketching

How many real values of $x$ satisfy the equation

$\large {x}^{2}-{2}^{x}=0?$

$6 \ln (x^2 + 1) - x = 0$

How many real solutions exist for the equation above?

Let $f(x)=x^4-6x^2+5.$ If $P(x_0,y_0)$ is a point such that $y_0>f(x_0)$ and there are exactly **two distinct tangents** drawn to the curve $y=f(x),$ what is the maximum value of $y_0?$

Try : Part-2

The function $f(x)= x^{3}-11x^{2} +19x+13$ has zeros $a_{1},a_{2},a_{3}.$ Then what is the value of $[ a_{1} ]+[ a_{2} ]+[ a_{3} ]?$

**Note:** $[m]$ Represents the greatest integer less than or equal to $m.$