Q1
At a point of local minimum, the second derivative test requires:
📝 SolutionA local minimum occurs at a critical point where $f''(x) > 0$ (the curve is concave up).
Q2
The slope of the tangent to the curve $y = x^2$ at the point $(2, 4)$ is:
📝 Solution$\frac{dy}{dx} = 2x$. At $x = 2$, slope $= 2(2) = 4$.
Q3
The equation of the normal to a curve at a point is:
📝 SolutionThe normal to a curve at a point is the line perpendicular to the tangent at that point.
Q4
If the slope of the tangent at a point is $m$, the slope of the normal at that point is:
📝 SolutionSince the normal is perpendicular to the tangent, the product of their slopes is $-1$, so the normal's slope is $-\frac{1}{m}$.
Q5
For a function to have a critical point at $x = a$, it is necessary that:
📝 SolutionCritical points occur where the derivative is zero or fails to exist, which are the candidates for local maxima or minima.
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Class 12 · Maths
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