Mathematics

Mathematics

Problem

Consider the quadratic function h(x)=kx2−4x+1h(x)=k x^{2}-4x+1, where kk is a real constant. For which value of kk does hh have a critical point at x=2x=2? a) k=12k=\dfrac12 b) k=74k=\dfrac74 c) k=1k=1 d) k=2k=2
Intuitive explanation

A critical point occurs where the derivative equals zero—that's where the function stops increasing or decreasing. Since hh is quadratic, its derivative is linear, and you can find it by differentiating term by term. Once you set that derivative equal to zero and use the condition that the critical point is at x=2x=2, you'll have a single equation in kk to solve.

Intuitive explanation

A critical point occurs where the derivative equals zero—that's where the function stops increasing or decreasing. Since hh is quadratic, its derivative is linear, and you can find it by differentiating term by term. Once you set that derivative equal to zero and use the condition that the critical point is at x=2x=2, you'll have a single equation in kk to solve.

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