For What Values Of $c$ Does the Curve Have Maximum and Minimum Points for the and Minimum

question 155

Multiple Choice

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$
Select the correct answer.

• A) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,2 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens downward with one minimum points.
• For \(c < 0\), the graph opens upward, and has an absolute maximum at \(x = 0\) and no local minimum.
• B) For \(c = 0 , f ( x ) = 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,2 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute minimum at \(x = 0\) and no local minimum.
• C) For \(c = 0 , f ( x ) = 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,3 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute minimum at \(x = 0\) and no local minimum.
• D) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,1 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute maximum at \(x = 0\) and no local minimum.
• E) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0 , - 1 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } - 1\), opens downward with two maximum points.
• For \(c < 0\), the graph opens upward, and has an absolute minimum at \(x = 0\).

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Definitions:

Genetically Influenced

Refers to traits or conditions that are affected by genes inherited from one's parents.

Prenatal Development

The process of growth and development within the womb from conception to birth, involving critical stages for physical and neurological formation.

Fetal Abnormalities

Physical or genetic anomalies that occur in the fetus during pregnancy, which can lead to various health issues or developmental disorders.

Deformities

are physical abnormalities or misshapen parts of the body, often present from birth or developed later in life, that deviate from the normal shape or size.